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All Textbook Solutions for Calculus Volume 3

For the following exercises, find a polar equation of theconic with focus at the origin and eccentricity and directrixas given. 287. Directrix: y=2;e=2 For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 288. Directrix: y=2;e=12 For the following exercises, sketch the graph of each conic. 289. r=11+sin For the following exercises, sketch the graph of each conic. 290. r=11cos For the following exercises, sketch the graph of each conic. 291. r=41+cos For the following exercises, sketch the graph of each conic. 292. r=105+4sin For the following exercises, sketch the graph of each conic. 293. r=1532cos For the following exercises, sketch the graph of each conic. 294. r=323+5sin For the following exercises, sketch the graph of each conic. 295. r(2+sin)=4 For the following exercises, sketch the graph of each conic. 296. r=32+6sin For the following exercises, sketch the graph of each conic. 297. r=34+2sin For the following exercises, sketch the graph of each conic. 298. x29+y24=1 For the following exercises, sketch the graph of each conic. 299. x24+y216=1 For the following exercises, sketch the graph of each conic. 300. 4x2+9y2=36 For the following exercises, sketch the graph of each conic. 301. 25x24y2=100 For the following exercises, sketch the graph of each conic. 302. x216y29=1 For the following exercises, sketch the graph of each conic. 303. x2=12y For the following exercises, sketch the graph of each conic. 304. y2=20x For the following exercises, sketch the graph of each conic. 305. 12x=5y2 For the following equations, determine which of the conic sections is described. 306. xy=4 For the following equations, determine which of the conic sections is described. 307. x2+4xy2y26=0 For the following equations, determine which of the conic sections is described. 308. x2+23xy+3y26=0 For the following equations, determine which of the conic sections is described. 309. x2xy+y22=0 For the following equations, determine which of the conic sections is described. 310. 34x224xy+41y225=0 For the following equations, determine which of the conic sections is described. 311. 52x272xy+73y2+40x+30y75=0 The mirror in an automobile headlight has a parabolic cross section, with the lightbulb at the focus. On a schematic, the equation of the parabola is given as x2=4y . At what coordinates should you place the lightbulb?A satellite dish is shaped like a paraboloid of revolution. The receiver is to be located at the forms. If the dish is 12feet across at its opening and 4feet deep at its center, where should the receiver be placed?Consider the satellite dish of the preceding problem. If the dish is 8feet across at the opening and 2feet deep, where should we place the receiver?A searchlight is shaped like a paraboloid of revolution. A light source is located 1foot from the base along the axis of symmetry. If the opening of the searchlight is 3feet across, find the depth.Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120feet and the foci are located 30feet from the center, find the height of the ceiling at the center.A person is standing 8feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80feet away, what is the length and the height at the center of the gallery?For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). 318. Halley’s Comet: length of major axis =35.88, eccentricity =0.967 For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). 319. Hale-Bopp Comet: length of major axis =525.91, eccentricity =0.995 For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). 320. Mars: length of major axis =3.049, eccentricity =0.0934 For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU). 321. Jupiter: length of major axis =10.408, eccentricity =0.0484 True or False? Justify your answer with a proof or a counter example. 322. The rectangular coordinates of the point (4,56) are (23,2)True or False? Justify your answer with a proof or a counter example. 323. The equations x=cosh(3t),y=2sinh(3t) represent a hyperbola.True or False? Justify your answer with a proof or a counter example. 324. The arc length of the spinal given by r=2 for 03 is 943 True or False? Justify your answer with a proof or a counter example. 325. Given x=f(t) and y=g(t), if dxdy=dydx, then f(t)=g(t)+C , where C is a constant.For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 326. x=1+t,y=t21,1t1 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 327. x=et,y=1e3t,0t1 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 328. x=sin,y=1csc,02 For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 329. x=4cos,y=1sin,02 For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. 330. r=4sin(3)For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. 331. r=5cos(5)For the following exercises, find the polar equation for the curve given as a Cartesian equation. 332. x+y=5 For the following exercises, find the polar equation for the curve given as a Cartesian equation. 333. y2=4+x2 For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. 334. x=ln(t),y=t21,t=1 For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. 335. r=3+cos(2),=34 For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. 336. Find dydx,dxdy, and d2xdy2 of y=(2+et) , x=1sin(t)For the following exercises, find the area of the region. 337. x=t2,y=ln(t),0te For the following exercises, find the area of the region. 338. r=1sin in the first quadrant For the following exercises, find the arc length of the curve over the given interval. 339. x=3t+4,y=9t2,0t3 For the following exercises, find the arc length of the curve over the given interval. 340. r=6cos,02 . Check your answer by geometry.For the following exercises, find the Cartesian equation describing the given shapes. 341. A parabola with focus (2,5) and directrix x=6 For the following exercises, find the Cartesian equation describing the given shapes. 342. An ellipse with a major axis length of 10 and foci at (7,2) and (1,2)For the following exercises, find the Cartesian equation describing the given shapes. 343. A hyperbola with vertices at (3,2) and (5,2) and foci at (2,6) and (2,4)For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. 344. r=61+3cos()For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. 345. r=432cos For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. 346. r=755cos Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26AU and miner axis is 38.07AU . What is the eccentricity?The C/1980E1 comet was observed in 1980 . Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364AU , find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? (Hint: Consider the Sun at point (0,0) .)For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 1. PQ For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 2. PR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 3. QP For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 4. RP For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 5. PQ+PR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 6. PQPR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 7. 2PQ2PR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 8. 2PQ+12PR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 9. The unit vector in the direction of PQ For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 10. The unit vector in the direction of PR For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 11. A vector v has initial point (1,3) and terminal point (2,1) . Find the unit vector in the direction of v. Express the answer in component form.For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 12. A vector v has initial point (2,5) and terminal point (3,1) . Find the unit vector in the direction of v. Express the answer in component form.For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 13. The vector v has initial point P(1,0) and terminal point Q that is on the y-axis and above the initial point. Find the coordinates of terminal point Q such that the magnitude 0f the vector v is 5 .For the following exercises, consider points P(1,3) , Q(1,5) , and R(3,7) . Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors. 14. The vector v has initial point P(1,1) and terminal point Q that is on the x-axis and left of the initial point. Find the coordinates of terminal point Q such that the magnitude of the vector v is 10 Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 15. a=2i+j,b=i+3j Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 16. a=2i,b=2i+2j Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 17. Let a be a standard-position vector with terminal point (2,4) . Let b be a vector with initial point (1,2) and terminal point (1,4) . Find the magnitude of vector 3a+b4i+j .Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 18. Let a be a standard-position vector with terminal point at (2,5) . Let b be a vector with initial point (1,3) and terminal point (1,0) . Find the magnitude of vector a3b+14i14j .Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 19. Let u and v be two nonzero vectors that are nonequivalent. Consider the vectors a=4u+5v and b=u+2v defined in terms of u and v. Find the scalar such that vectors a+b and uv are equivalent.Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 20. Let u and v be two nonzero vectors that are nonequivalent. Consider the vectors a=2u4v and b=3u7v defined in terms of u and v. Find the scalars and such that vectors a+b and uv are equivalent.Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 21. Consider the vector a(t)=cost,sint with components that depend on a real number t. As the number t varies, the components of a(t) change as well, depending on the functions that define them. a. Write the vectors a(0) and a() in component form. b. Show that the magnitude a(t) of vector a(t) remains constant for any real number t. c. As t varies, show that the terminal point of vector a(t) describes a circle centered at the origin of radius 1.Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 22. Consider vector a(x)=x,1x2 with components that depend on a real number x[1,1] . As the number x varies, the components of a(x) change as well, depending on the functions that define them. a. Write the vectors a(0) and a(1) in component form. b. Show that the magnitude a(x) of vector a(x) remains constant for any real number x. c. As x varies, show that the terminal point of vector a(x) describes a circle centered at the origin of radius 1.Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 23. Show that vectors a(t)=cot,sint and a(x)=x.1x2 are equivalent for x=r and t=2k , where k is an integer.Far the following exercises, use the given vectors. a and b. Determine the vector sum a+b and express it in both the component form and by using the standard unit vectors Find the vector difference ab and express it in both the component form and by using the standard unit vectors. Verify that the vectors a, b, and a+b , and, respectively, a, b, and ab satisfy the triangle inequality Determine the vectors 2a , b , and 2ab . Express the vectors in both the component form and by using standard unit vectors. 24. Show that vectors a(t)=cost,sint and a(x)=x,1x2 are opposite for x=r and t=2kr where k is an integer.For the following exercises, find vector v with the given magnitude and in the same direction as vector u. 25. v=7,u=3,4 For the following exercises, find vector v with the given magnitude and in the same direction as vector u. 26. v=3,u,2,5 For the following exercises, find vector v with the given magnitude and in the same direction as vector u. 27. v=7,u=3,5 For the following exercises, find vector v with the given magnitude and in the same direction as vector u. 28. v=10,u=2,1 For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 29. u=2,=30 For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 30. u=6,=60 For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 31. u=5,=2 For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 32. u=8,=For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 33. u=10,=56 For the following exercises, find the component form of vector u, given its magnitude and the angle the vector makes with the positive x-axis. Give Exam answers when possible. 34. u=50,=34 For the following exercises, vector u is given. Find the angle [0,2) that vector u makes with the positive direction of the x-axis, in a counter-clockwise direction. 35. u=52i52j For the following exercises, vector u is given. Find the angle [0,2) that vector u makes with the positive direction of the x-axis, in a counter-clockwise direction. 36. u=3ij Let a=a1,a2,b=b1,b2 , and c=c1,c2 be three nonzero vectors. If a1b2a2b10 , then show there are two scalars, and , Such that c=a+b .Consider vectors a=2,4,b=1,2, and c=0 . Determine the scalars and such that c=a+b.Let P(x0,f(x0)) be a fixed point on the graph of the differential function f with a domain that is the set of real numbers. Determine the real number z0 such that point Q(x0+1,z0) is situated on the line tangent to the graph of f at point P . Determine the unit vector u with initial point P and terminal point Q .Consider the function f(x)=x4 , where x. Determine the real number z0 Such that point Q(2,z0) situated on the line tangent to the graph of f at point P(1,1) Determine the unit vector u with initial point P and terminal point Q .Consider f and g two functions defined on the same set of real numbers D . Let a=x,f(x) and b=x,g(x) be two vectors that describe the graphs of the functions, where xD. Show that if the graphs of the functions f and g do not intersect, then the vectors a and b are not equivalent.Find x such that vectors a=x,sinx and b=x,cosx are equivalent.Calculate the coordinates of point D such that ABCD is a parallelogram, with A(1,1),B(2,4), and C(7,4) .Consider the points A(2,1),B(10,6),C(13,4) , and D(16,2) . Determine the component form of vector AD .The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of 70mph and an angle of elevation of 30 . Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)A baseball player throws a baseball at an angle of 30 with the horizontal. If the initial speed of the hall is 100mph , find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)A bullet is fired with an initial velocity of 1500ft/sec at an angle of 60 with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)[T] A 65kg sprinter exerts a force of 798N at a 19 angle with respect to the ground 011 the starting block at the instant a race begins. Find the horizontal component of the force. [Round to two decimal places.)[T] Two forces, a horizontal force of 45lb and another of 52lb , act on the same object. The angle between these forces is 25 . Find the magnitude and direction angle from the positive x-axis 0f the resultant force that acts on the object. (Round to two decimal places.)[T] Two fumes, a vertical fence of 26lb and another of 45lb , act on the same object. The angle between these forces is 55 . Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)[T] Three forces act an object. Two of the fences have the magnitudes 58N and 27N , and make angles 53 and 152 , respectively, with the positive x-axis . Find the magnitude and the direction angle from the positive x-axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)Three forces with magnitudes 80lb,120lb, and 60lb act on an abject at angles of 45,60 and 30 , respectively, with the positive x-axis . Find the magnitude and direction angle from the positive x-axis of the resultant farce. (Round to two decimal places.)[T] An airplane is flying in the direction of 43 east of north (also abbreviated as N43E ) at a speed of 550mph . A wind with speed 25mph comes from the southwest at a hearing of N15E . What are the ground speed and new direction of the airplane?[T] A boat is traveling in the water at 30mph in a direction of N20E (that is, 20 east of north). A strong current is moving at 15mph in a direction of N45E . What are the new speed and direction of the boat?[T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of 40 and 53 ,respectively, with the horizontal. Find the magnitudes of the forces of tension T1 and T2 in the cables if the resultant fame acting on the object is zero. (Round to two decimal places.)[T] A 62-lh weight hangs from a nope that makes the angles of 29 and 61 , respectively, with the horizontal. Find the magnitudes of the forces of tension T1 and T2 in the cables if the resultant force acting on the object is zero. (Round to two decimal planes.)[T] A 1500lb boat is parked on a ramp that makes an angle of 30 with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector v1 that is parallel to the ramp and a vertical vector v2 that is perpendicular to the inclined surface. The magnitudes of vectors v1 and v2 are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of v1 and v2 . (Round to the nearest integer.)[T] An 85lb box is at rest on a 26 incline. Determine the magnitude of the fame parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)A guy-wire supports a pole that is 75ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground 50ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is 50lb . (Round to the nearest integer.)A telephone pole guy-wire has an angle of elevation of 35 with respect to the ground. The force of tension in the guy-wire is 120lb . Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point A(2,3,5) is the opposite vertex to the origin, then find a. the coordinates of the other six vertices of the box and b. the length of the diagonal of the box determined by the vertices O and A .Find the coordinates of point P and determine its distance to the origin.For the following exercises, describe and graph the set of points that satisfles the given equation. 63. (y5)(z6)=0 For the following exercises, describe and graph the set of points that satisfles the given equation. 64. (z2)(z5)=0 For the following exercises, describe and graph the set of points that satisfles the given equation. 65. (y1)2+(z1)2=1 For the following exercises, describe and graph the set of points that satisfles the given equation. 66. (x2)2+(z5)2=4 Write the equation of the plane passing through point (1,1,1) that is parallel to the xy -plane.Write the equation of the plane passing through point (1,3,2) that is parallel to the xz -plane.Find an equation of the plane passing through points (1,3,2),(0,3,2) and (1,0,2) .Find an equation of the plane passing through points (1,9,2),(1,3,6), and (1,7,8) .For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions. 71. Center C(1,7,4) and radius 4 For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions. 72. Center C(4,7,2) and radius 6 For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions. 73. Diameter PQ, where P(1,5,7) and Q(5,2,9)For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions. 74. Diameter PQ , where P(16,3,9) and Q(2,3,5)For the following exercises, find the center and radius of the sphere with an equation in general form that is given. 75. P(1,2,3)x2+y2+z24z+3=0 For the following exercises, find the center and radius of the sphere with an equation in general form that is given. 76. x2+y2+z26x+8y10z+25=0 For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 77. P(3,0,2) and Q(1,1,4)For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 78. P(0,10,5) and Q(1,1,3)For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 79. P(2,5,8) and M(1,7,4) , where M is the midpoint of the line segment PQ For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 80. Q(0,7,6) and M(1,3,2) , where M is the midpoint of the Line segment PQ For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 81. Find terminal point Q of vector PQ=7,1,3 with the initial point at P(2,3,5) .For the following exercises, express vector PQ with the initial paint at P and the terminal point at Q a. in component form and b. by using standard unit vectors. 82. Find initial point P of vector PQ=9,1,2 with the terminal point at Q(10,0,1) .For the following exercises, use the given vectors a and b to find and express the vectors a+b,4a , and -5a+3b in component form. 83. a=1,2,4,b=5,6,7 For the following exercises, use the given vectors a and b to find and express the vectors a+b,4a , and -5a+3b in component form. 84. a=3,-2,4,b=-5,6,-9 For the following exercises, use the given vectors a and b to find and express the vectors a+b,4a , and -5a+3b in component form. 85. a=-k,b=-i For the following exercises, use the given vectors a and b to find and express the vectors a+b,4a , and -5a+3b in component form. 86. a=i+j+k,b=2i3j+2k For the following exercises, vectors u and v are given. Find the magnitudes of vectors u-v and -2u . 87. u=2i+3j+4k,v=i+5jk For the following exercises, vectors u and v are given. Find the magnitudes of vectors u-v and -2u . 88. u=i+j,v=j-k For the following exercises, vectors u and v are given. Find the magnitudes of vectors u-v and -2u . 89. u=2cost,2sint,3,v=0,0,3, where t is a real number.For the following exercises, vectors u and v are given. Find the magnitudes of vectors u-v and -2u . 90. u=0,1,sinht,v=1,1,0 , where t is a real number.For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 91. a=3i4j For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 92. a=4,3,6 For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 93. a=PQ, where P(2,3,1) and Q(0,4,4) .For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 94. a=OP , where P(1,1,1)For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 95. a=uv+w, where u=1jk,v=2ij+k, and w=i+j+3k For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors. 96. a=2u+vw, where u=ik,v=2j, and w=ij Determine whether AB and PQ are equivalent vectors, where A(1,1,1),B(3,3,3),P(1,4,5), and Q(3,6,7) .Determine whether the vectors AB and PQ are equivalent, where A(1,4,1),B(2,2,0),P(2,5,7), and Q(3,2,1) .For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 99. v=7,1,3,u=10,u and v have the same direction For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 100. v=2,4,1,u=15,u and v have the same direction For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 101. v=2sint,2cost,1,u=2,u and v have opposite directions for any t , Where t is a real number For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions. 102. v=3sinht,0,3,u=5,u and v have opposite directions for any t , where t is a real number Determine a vector of magnitude 5 in the direction of vector AB , where A(2,1,5) and B(3,4,-7) .Find a vector of magnitude 2 that points in the opposite direction than vector AB , where A(1,1,1) and B(0,1,1) . Express the answer in component form.Consider the points A(2,,0),B(0,1,), and C(1,1,), where and are negative real numbers. Find and such that OAOB+OC=OB=4 .Consider points A(,0,0),B(0,,0), and C(,,), where and are positive real numbers. Find and such that OA+OB=2 , find OC=3 .Let P(x,y,z) be a point situated at an equal distance from points A(1,1,0) and B(1,2,1) . Show that paint P lies on the plane of equation 2x+3y+z=2 .Let P(x,y,z) be a point situated at an equal distance from the origin and point A(4,1,2) . Show that the coordinates of point P satisfy the equation 8x+2y+4z=21 .The points A,B, and C are collinear (in this order) if the relation AB+BC=AC is satisfied. Show that A(5,3,1),B(5,3,1) , and C(15,9,3) are collinear points.Show that points A(1,0,1),B(0,1,1) and C(1,1,1) are not collinear.[T] A force F of 50N acts on a particle in the direction of the vector OP , where P(3,4,0) . a. Express the force as a vector in component form. b. Find the angle between force F and the positive direction of the x -axis. Express the answer in degrees rounded to the nearest integer.[T] A force F of 40N acts on a box in the direction of the vector OP , where P(1,0,2) . a. Express the fence as a vector by using standard unit vectors. b. Find the angle between force F and the positive direction of the x -axis.If F is a force that moves an object from point P1(x1,y1,z1) to another point P2(x2,y2,z2) , then the displacement vector is defined as D=(x2x1)i+(y2y1)j+(z2z1)k. A metal container is lifted 10m vertically by a constant force F . Express the displacement vector D by using standard unit vectors .A box is pulled 4 yd horizontally in the x -direction by a constant force F . Find the displacement vector in component form.The sum of the forces acting on an object is called the resultant or net force. All abject is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let F1=10,6,3,F2=0,4,9, and F3=10,3,9 be three forces acting on a box. Find the force F4 acting on the box such that the box is in static equilibrium. Express the answer in component form.[T] Let Fk=1,k,k2,k=1,...,n be n forces acting on a particle, with n2 . a. Find the net force F=k=1nFk . Express the answer using standard unit vectors. b. Use a computer algebra system (CA5) to find n such that F100 .The force of gravity F acting on an object is given by F=mg, where m is the mass of the object (expressed in kilograms) and g is acceleration resulting from gravity, with g=9.8N/kg. A 2kg disco ball hangs by a Chain from the ceiling of a room. Find the force of gravity Facting on the disco ball and find its magnitude. Find the force of tension T in the chain and its magnitude. Express the answers using standard unit vectors. Figure 2.43 (credit: modification of work by Kenneth Lu, Flickr)A 5-kg pendant chandelier is designed such that the alabaster bowl is held by four chains of equal length, as shown in the following ?gure. Find the magnitude of the force of gravity acting on the chandelier. Find the magnitudes of the forces of tension for each of the four chains (assume chains are essentially vertical).[T] A 30-kg block of cement is suspended by three cables of equal length that are anchored at points P(2,0,0),Q(1,3,0) and R(1,3,0) . The load is located at S(0,0,23), as shown in the following ?gure. Let F1,F2, and F3 be the forces of tension resulting from the load in tables RS,QS, and PS, respectively. Find the gravitational force F acting on the block of cement that counterbalances the sum F1+F2+F3 of the forces of tension in the cables. Find forces F1,F2, and F3 . Express the answer in component form.Two soccer players are practicing for an upcoming game. One of them runs 10m from point A to point B . She then turns left at 90 and runs 10m until she reaches point C. Then she kicks the hall with a speed of 10m/sec at an upward angle of 45 to her teammate, who is located at point A . Write the velocity 0f the ball in component form.Let r(t)=x(t),y(t),z(t) be the position vector of a panicle at the time t[0,T], Where x,y, and z are smooth functions on [0,T]. The instantaneous velocity of the particle at time t is de?ned by vector v(t)=x(t),y(t)z(t), with components that are the derivatives with respect to t, of the functions x,y, and z, respectively. The magnitude v(t) of the instantaneous velocity vector is called the speed of the particle at time t. Vector a(t)=x(t),y(t),z(t), with components that are the second derivatives with respect to t, of the functions x,y, and z, respectively, gives the acceleration of the particle at time t. Consider r(t)=cost,sint,2t the position vector of a particle at time t[0,30], where the components of r are expressed in centimeters and time is expressed in seconds. Find the instantaneous velocity, speed, and acceleration of the particle after the first second. Round your answer to two decimal places. Use a CAS to visualize the path of the particle—that is, the set of all points of coordinates(cost,sint,2t) where t[0,30] .[T] Let r(t)=t,2t2,4t2 be the position vector of a particle at time t (in seconds), where t[0,10] (here the components of r are expressed in centimeters). Find the instantaneous velocity, speed, and acceleration of the particle after the first two seconds. Round your answer to two decimal places. Use a CAB to visualize the path of the particle de?ned by the points (t,2t2,4t2) where t[0,60].For the following exercises, the vectors u and v are given. Calculate the dot product uv . 123. u=3,0,v=2,2 For the following exercises, the vectors u and v are given. Calculate the dot product uv . 124. u=3,4,v=4,3 For the following exercises, the vectors u and v are given. Calculate the dot product uv . 125. u=2,2,1,v=4,3 For the following exercises, the vectors u and v are given. Calculate the dot product uv . 126. u=4,5,6,v=0,2,3 For the following exercises, the vectors a,b, and c are given. Determine the vectors (ab)c and (ac)b . Express the vectors in component form. 127. a=2,0,3,b=4,7,1,c=1,1,1 For the following exercises, the vectors a,b, and c are given. Determine the vectors (ab)c and (ac)b . Express the vectors in component form. 128. a=0,1,2,b=1,0,1,c=1,0,1 For the following exercises, the vectors a,b, and c are given. Determine the vectors (ab)c and (ac)b . Express the vectors in component form. 129. a=i+j,b=ik,c=i2k For the following exercises, the vectors a,b, and c are given. Determine the vectors (ab)c and (ac)b . Express the vectors in component form. 130. a=i-j+k,b=j+3k,c=-i+2j-4k For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle between a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is an acute angle? 131. [T]a=3,1,b=(4,0)For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle between a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is an acute angle? 132. [T]a=2,1,b=1,3 For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle between a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is an acute angle? 133. u=3i,v=4i+4j For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle between a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is an acute angle? 134. u=5i,v=6i+6j For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 135. a=3,1,2,b=1,1,2 For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 136. a=0,1,3,b=2,3,1 For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 137. a=i+j,b=jk For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 138. a=i2j+k,b=i+j2k For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 139. [T]a=3ij2k,b=v+w, where v=2i3j+2k and w=i+2k For the following exercises, find the measure 0f the angle between the three-dimensional vectors a and b . Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. 140. [T]a=3ij+2k,b=vw, where v=2i+j+4k and w=6i+j+2k For the following exercises determine whether the given vectors are orthogonal. 141. a=x,y,z,b=y,x, where x and y are nonzero real numbers For the following exercises determine whether the given vectors are orthogonal. 142. a=x,x,b=y,y, where x and y are nonzero real numbers For the following exercises determine whether the given vectors are orthogonal. 143. a=3ij2k,b=2i3j+k For the following exercises determine whether the given vectors are orthogonal. 144. a=ij,b=7i+2jk For the following exercises determine whether the given vectors are orthogonal. 145. Find all two-dimensional vectors a orthogonal to vector b=3,4 . Express the answer in component form.For the following exercises determine whether the given vectors are orthogonal. 146. Find all two-dimensional vectors a orthogonal to vector b=5,6 . Express the answer by using standard unit vectors.For the following exercises determine whether the given vectors are orthogonal. 147. Determine all three-dimensional vectors u orthogonal to vector v=1,1,0 . Express the answer by using standard unit vectors.For the following exercises determine whether the given vectors are orthogonal. 148. Determine all three-dimensional vectors u orthogonal to vector v=ijk . Express the answer in component form.For the following exercises determine whether the given vectors are orthogonal. 149. Determine the real number a such that vectors a=2i+3j and b=9i+aj are orthogonal.For the following exercises determine whether the given vectors are orthogonal. 150. Determine the real number a such that vectors a=3i+2j and b=2i+aj are orthogonal.[T] Consider the points P(4,5) and Q(5,7) . Determine vectors OP and OQ . Express the answer by using standard unit vectors. Determine the measure of angle O in triangle OPQ . Express the answer in degrees rounded to two decimal places.[T] Consider points A(1,1),B(2,7) , and C(6,3). Determine vector BA and BC . Express the answer in component form. Determine the measure of angle B in triangle ABC . Express the answer in degrees rounded to two decimal places.Determine the measure of angle A in triangle ABC , where A(1,1,8),B(4,3,4), and C(3,1,5). Express your answer in degrees rounded to two decimal places.Consider points P(3,7,2) and Q(1,1,3). Determine the angle between vectors OP and OQ . Express the answer in degrees rounded to two decimal places.For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 155. u=3,7,2,v=5,3,3,w=0,1,1 For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 156. u=ik,v=5j5k,w=10j For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 157. Use vectors to show that a parallelogram with equal diagonals is a square.For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 158. Use vectors to show that the diagonals of a rhombus are perpendicular.For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 159. Show that u(v+w)=uv+uw is true for any vector u,v, and w.For the following exercises, determine which (if any) pairs of the following vectors are orthogonal. 160. Verify the identity u(v+w)=uv+uw for vectors u=1,0,14,v=2,3,5, and w=4,2,6.For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 161. u=2,2,1 For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 162. u=i2j+2k For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 163. u=1,5,2 For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 164. u=2,3,4 For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 165. Consider u=a,b,c a nonzero three- dimensional vector. Let cos,cos, and cos be the directions 6f the cosines of u . Show that cos2+cos2+cos2=1 .For the following problems, the vector u is given. Find the direction cosines for the vector u . Find the direction angles for the vector u expressed in degrees. {Round the answer to the nearest integer.) 166. Determine the direction cosines of vector u=i+2j+2k and show they satisfy cos2+cos2+cos2=1 .For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 167. u=5i+2j,v=2i+3j For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 168. u=4,7,v=3,5 For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 169. u=3i+2k,v=2j+4k For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 170. u=4,4,0,v=0,4,1 For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 171. Consider the vectors u=4i3j and v=3i+2j . Find the component form of vector w=projuv that represents the projection of v onto u . Write the decomposition v=w+q of vector v into the orthogonal components w and q , where w is the projection of v onto u and q is a vector orthogonal to the direction of u .For the following exercises, the vectors u and v are given. Find the vector projection w=projuv of vector v onto vector u . Express your answer in component form. Find the scalar projection compuv of vector v onto vector u .. 172. Consider vectors u=2i+4j and v=4j+2k . Find the component form of vector w=projuv that represents the projection of v unto u . Write the decomposition v=w+q of vector v into the orthogonal components w and q , where w is the projection of v onto u and q is a vector orthogonal to the direction of u .A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points P(1,1,1),Q(1,1,1),R(1,1,1) and S(1,1,1) (see figure). Find the distance between the hydrangea atoms located at P and R . Find the angle between vectors OS and OR that connect the carbon atom with the hydrogen atoms located at S and R, which is also called the bond angle. Express the answer in degrees rounded to two decimal places.[T] Find the vectors that join the center of a clock to the hours 1:00,2:00, and 3:00. Assume the clock is circular with a radius of 1unit .Find the work done by force F=5,6,2 (measured in Newtons) that moves a particle form point P(3,1,0) to point Q(2,3,1) along a straight line (the distance is measured in meters).[T] A sled is pulled by exerting a force of 100N on a rope that makes an angle of 25 with the horizontal. Find the work done in pulling the sled 40m . (Round the answer to one decimal place.)[T] A father is pulling his son on a sled at an angle of 20 with the horizontal with a force of 25lb (see the following image). He pulls the sled in a straight path of 50ft . How much work was done by the man pulling the sled? (Round the answer to the nearest integer.)[T] A car is towed using a force of 1600N . The rope used to pull the car makes an angle of 25 with the horizontal. Find the work done in towing the car 2km . Express the answer in joules (1J=1Nm) rounded to the nearest integer.[T] A boat sails north aided by a wind blowing in a direction of N30E with a magnitude of 500lb . Haw much work is performed by the wind as the boat moves 100ft ? (Round the answer to two decimal places.)Vector p=150,225,375 represents the price of certain models of bicycles sold by a bicycle shop. Vector n=10,7,9 represents the number of bicycles sold of each model, respectively. Compute the dot product pn and state its meaning.[T] Two forces F1 and F2 are represented by vectors with initial points that are at the origin. The first farce has a magnitude of 20lb and the terminal point of the vector is point P(1,1,0). The second force has a magnitude of 40lb and the terminal point of its vector is point Q(0,1,1). Let F be the resultant fame of forces F1 and F2. Find the magnitude of F . (Round the answer to one decimal place.) Find the direction angles of F . (Express the answer in degrees rounded to one decimal place.)[T] Consider r(t)=cost,sint,2t the position vector of a particle at time t[0,30] , where the components of r are expressed in centimeters and lime in seconds. Let OP be the position vector of the particle after 1sec . Show that all vector PQ , where Q(x,y,z) is an arbitrary point, orthogonal to the instantaneous velocity vector v(1) of the particle after 1sec , can he expressed as PQ=xcos1,ysin1,z2, where xsin1ycos12z+4=0. The set of point Q describes a plane called the normal plane to the path of the particle at point P . Use a CAB to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 183. u=2,0,0,v=2,2,0 For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 184. u=3,2,1,v=1,1,0 For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 185. u=2i+3j,v=j+2k For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 186. u=2j+3k,v=3i+k `For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 187. Simplify (ii2ij4ik+3jk)i .For the following exercises, the vectors u and v are given. Find the cross product uv of the vectors u and v . Express the answer in component form. Sketch the vectors u,v , and uv. 188. Simplify j(kj+2j+2ji3jj+5ik) .In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 189. u=3,1,2,v=2,0,1 In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 190. u=2,6,1,v=3,0,1 In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 191. u=AB,v=AC, where A(1,0,1),B(1,1,3), and C(0,0,5)In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 192. u=OP,v=PQ , where P(1,1,0) and Q(0,2,1)In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 193. Determine the real number such that uv and i are orthogonal, where u=3i+j5k and v=4i2j+k .In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 194. Show that uv and 2i14j+2k cannot be orthogonal for any a real number, where u=i+7jk and v=i+5j+k.In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 195. Show that uv is orthogonal to u+v and uv , where u and v are nonzero vectors.In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector uv . Express your answer using standard unit vectors. 196. Show that vu is orthogonal to (uv)(u+v)+u , when; u and v are nonzero vectors.Calculate the determinant |ijk117203| .Calculate the determinant |ijk034161| .For the following exercises, the vector u and v are given. Use determinant mutation to find vector w orthogonal to vectors u and v . 199. u=1,0,et,v=1,et,0 , where t is a real number For the following exercises, the vector u and v are given. Use determinant mutation to find vector w orthogonal to vectors u and v . 200. u=1,0,x,v=2x,1,0, where x is a nonzero real number Find vector (a2b)c , where a=|ijk215018|,b=|ijk011212|, and c=i+j+k .Find vector c(a+3b) , where a=|ijk509090|,b=|ijk011711|, and c=ik.[T] Use the cross product uv to find the acute angle between vectors u and v, where u=i+2j and v=i+k. Express the answer in degrees rounded to the nearest integer.[T] Use the cross product uv to find the obtuse angle between vectors u and v , where u=i+3j+k and v=i2j . Express the answer in degrees rounded to the nearest integer.Use the sine and cosine of the angle between two nonzero vectors u and v to prove Lagrange’s identity: uv2=u2v2(uv)2 Verify Lagrange’s identity uv2=u2v2(uv)2 for vectors u=i+j2k and v=2ij Nonzero vectors u and v are called collinear if there exists a nonzero scalar such that v=u. Show that u and v are collinear if and only if uv=0 .Nonzero vectors u and v are called collinear if there exists a nonzero scalar a such that v=u. Show that vectors AB and AC are collinear, where A(4,1,0),B(6,5,2), and C(5,3,1) .Find the area of the parallelogram with adjacent sides u=3,2,0 and v=0,2,1 .Find the area of the parallelogram with adjacent sides u=i+j and v=i+k .Consider paints A(3,1,2),B(2,1,5), and C(1,2,2). Find the area of parallelogram ABCD with adjacent sides AB and AC . Find the area of triangle ABC . Find the distance from point A to line BC .Consider points A(2,3,4),B(0,1,2), and C(1,2,0). Find the area of parallelogram ABCD with adjacent sides AB and AC . Find the area of triangle ABC . Find the distance from paint B to line AC .In the following exercises, vectors u,v, and w are given. Find the triple scalar product u(vw) Find the volume of the parallelepiped with the adjacent edges u,v, and w . 213. u=i+j,v=j+k, and w=i+k In the following exercises, vectors u,v, and w are given. Find the triple scalar product u(vw) Find the volume of the parallelepiped with the adjacent edges u,v, and w . 214. u=3,5,1,v=0,2,2, and w=3,1,1 In the following exercises, vectors u,v, and w are given. Find the triple scalar product u(vw) Find the volume of the parallelepiped with the adjacent edges u,v, and w . 215. Calculate the triple scalar products if v(uw) and w(uv), where u=1,1,1,v=7,6,9, and w=4,2,7.In the following exercises, vectors u,v, and w are given. Find the triple scalar product u(vw) Find the volume of the parallelepiped with the adjacent edges u,v, and w . 216. Calculate the triple scalar products w(vu) and u(wv), where u=4,2,1,v=2,5,3, and w=9,5,10.Find vectors a,b, and c with a triple scalar product given by the determinant |123025892|. Determine their triple scalar product.The triple scalar product of vector a,b, and c is given by the determinant |021014137|. Find vector ab+c .Consider the parallelepiped with edges OA,OB, and OC, where A(2,1,0),B(1,2,0), and C(0,1,). Find the real number 0 such that the volume of the parallelepiped is 3units3. For =1, find the height h from vertex C of the parallelepiped. Sketch the parallelepiped.Consider points A(,0,0),B(0,,0), and C(0,0,), with ,, and positive real numbers. Determine the volume of the parallelepiped with adjacent sides OA,OB, and OC. Find the volume of the tetrahedron with vertices O,A,B, and C. (Hint: The volume of the tetrahedron is 1/6 of the volume of the parallelepiped.) Find the distance from the origin to the plane determined by A,B, and C. Sketch the parallelepiped and tetrahedron.Let u,v, and w be three-dimensional vectors and c be a real number. Prove the following properties of the cross product. uu=0 u(v+w)=(uv)+(uw) c(uv)=(cu)v=u(cv) u(uv)=0 Show that vectors u=1,0,8,v=0,1,6, and w=1,9,3 satisfy the following properties of the cross product. uu=0u(v+w)=(uv)+(uw)c(uv)=(cu)v=u(cv)u(uv)=0 Nonzero vectors u,v, and w are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers and such that w=u+v. Otherwise, the vectors are called linearly independent. Show that u,v, and w are coplanar if and only if they are Linear dependent.Consider vectors u=1,4,7,v=2,1,4,w=0,9,18, and p=0,9,17. Show that u,v, and w are coplanar by using their triple scalar product Show that u,v, and w are coplanar, using the definition that there exist two nonzero real numbers and such that w=u+v. Show that u,v, and p are linearly independent—that is, none of the vectors is a linear combination of the other two.Consider points A(0,0,2),B(1,0,2),C(1,1,2), and D(0,1,2). Are vectors AB,AC, and AD linearly dependent (that is, one of the vectors is a linear combination of the other two)?Show that vectors i+j,ij, and i+j+k are linearly independent—that is, there exist two nonzero real numbers and such that i+j+k=(i+j)+(ij) .Let u=u1,u2,0 and v=v1,v2,0, be two-dimensional vector. The cross product of vectors u and v is net defined. However, if the vectors are regarded as the three-dimensional vectors u=u1,u2,0 and v=v1,v2,0, respectively, then, in this case, we can define the cross product of u and v . In particular, in determinant notation, the cross product of u and v is given by uv=|ijk u 1 u 20 v 1 v 20|. Use this result to compute (icos+jsin)(isinjcos) , where is a real number.Consider points P(2,1),Q(4,2) and R(1,2). Find the area of triangle P,Q, and R. Determine the distance from point R to the line passing through P and Q .Determine a vector of magnitude 10 perpendicular to the plane passing through the x -axis and point P(1,2,4).Determine a unit vector perpendicular to the plane passing through the z -axis and point A(3,1,2).Consider u and v two three-dimensional vectors. If the magnitude of the cross product vector uv is k times larger than the magnitude of vector u , show that the magnitude of v is greater than or equal to k , where k is a natural number.[T] Assume that the magnitudes of two nonzero vectors u and v are known. The function f()=uvsin defines the magnitude of the cross product vector uv, where [0,] is the angle between u and v . Graph the function f. Find the absolute minimum and maximum of function f. Interpret the results. If u=5 and v=2, find the angle between u and v if the magnitude of their cross product vector is equal to 9 .Find all vectors w=w1,w2,w3 that satisfy the equation 1,1,1w=1,1,2.- Solve the w1,0,1=3,0,3, where w=w1,w2,w3 is a nonzero vector with a equation magnitude of 3 .[T] A mechanic uses a 12in . wrench to turn a bolt. The wrench makes a 30 angle with the horizontal. If the mechanic applies a vertical force of 10lb on the wrench handle, what is the magnitude of the torque at point P (see the following figure}? Express the answer in foot-pounds rounded to two decimal places.[T] A boy applies the brakes on a bicycle by applying a downward force of 20lb on the pedal when the 6-in . crank makes a 40 angle with the horizontal (see the following figure). Find the torque at point P . Express your answer in foot-pounds rounded to two decimal places.[T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y -axis if the force is applied in the direction 0,1,2 and it produces a 100Nm torque to the bolt located at the origin.[T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of 35 to produce a torque of 20Nm ?

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