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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. 58. (x + 1)(y 3) = 0Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. 59. x2y2z2 0Sets of points Describe with a sketch the sets of points (x, y, z) satisfying the following equations. 60. y z = 0Sets of points 61. Give a geometric description of the set of points (x, y, z) satisfying the pair of equations z = 0 and x2 + y2 = 1. Sketch a figure of this set of points.Sets of points 62. Give a geometric description of the set of points (x, y, z) satisfying the pair of equations z = x2 and y = 0. Sketch a figure of this set of points.Sets of points 63. Give a geometric description of the set of points (x, y, z) that lie on the intersection of the sphere x2 + y2 + z2 = 5 and the plane z = 1.Sets of points 64. Give a geometric description of the set of points (x, y, z) that lie on the intersection of the sphere x2 + y2 + z2 = 36 and the plane z = 6.65E66EWrite the vector v = 2, 4, 4 as a product of its magnitude and a unit vector with the same direction as v.Find the vector of length 10 with the same direction as w = 2,2,3.Find a vector of length 5 in the direction opposite that of 3,2,3 .70E71EParallel vectors of varying lengths Find vectors parallel to v of the given length. 70. v=PQ with P(1, 0, 1) and Q(2, 1, 1); length = 3Parallel vectors of varying lengths Find vectors parallel to v of the given length. 69. v=PQ with P(3, 4, 0) and Q(2, 3, 1); length = 3Collinear points Determine the values of x and y such that the points (1, 2, 3), (4, 7, 1), and (x, y, 2) are collinear (lie on a line).Collinear points Determine whether the points P, Q, and R are collinear (lie on a line) by comparing PQ and PR. If the points are collinear, determine which point lies between the other two points. a. P(1, 6, 5), Q(2, 5, 3), R(4, 3, 1) b. P(1, 5, 7), Q(5, 13, 1), R(0, 3, 9) c. P(1, 2, 3), Q(2, 3, 6), R(3, 1, 9) d. P(9, 5, 1), Q(11, 18, 4), R(6, 3, 0)Lengths of the diagonals of a box What is the longest diagonal of a rectangular 2 ft 3 ft 4 ft box?Three-cable load A 500-kg load hangs from three cables of equal length that are anchored at the points (2, 0, 0), (1,3,0), and (1,3,0). The load is located at (0,0,23). Find the vectors describing the forces on the cables due to the load.Four-cable load A 500-lb load hangs from four cables of equal length that are anchored at the points (2, 0, 0) and (0, 2, 0). The load is located at (0, 0, 4). Find the vectors describing the forces on the cables due to the load.Possible parallelograms The points O(0, 0, 0), P(1, 4, 6), and Q(2, 4, 3) lie at three vertices of a parallelogram. Find all possible locations of the fourth vertex.80EMidpoint formula Prove that the midpoint of the line segment joining P(x1, y1, z1) and Q(x2, y2, z2) is (x1+x22,y1+y22,z1+z22).Equation of a sphere For constants a, b, c, and d, show that the equation x2+y2+z22ax2by2cz=d describes a sphere centered at (a, b, c) with radius r, where r2 = d + a2 + b2 + c2, provided d + a2 + b2 + c2 0.83EMedians of a trianglewith coordinates In contrast to the proof in Exercise 81, we now use coordinates and position vectors to prove the same result. Without loss of generality, let P(x1, y1, 0) and Q(x2, y2, 0) be two points in the xy-plane and let R(x3, y3, z3) be a third point, such that P, Q, and R do not lie on a line. Consider PQR. a. Let M1 be the midpoint of the side PQ. Find the coordinates of M1 and the components of the vector RM1. b. Find the vector OZ1from the origin to the point Z1 two-thirds of the way along RM1. c. Repeat the calculation of part (b) with the midpoint M2, of RQ and the vector PM2 to obtain the vector OZ2. d. Repeat the calculation of part (b) with the midpoint M3 of PR and the vector QM3 to obtain the vector OZ3. e. Conclude that the medians of PQR intersect at a point. Give the coordinates of the point. f. With P(2, 4, 0), Q(4, 1, 0), and R(6, 3, 4), find the point at which the medians of PQR intersect.The amazing quadrilateral propertycoordinate free The points P, Q, R, and S, joined by the vectors u, v, w, and x, are the vertices of a quadrilateral in 3. The four points neednt lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that u + v = w + x. b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = (u + v)/2. c. Let n be the vector that joins the midpoints of PS and SR. Show that n = (x + w)/2. d. Combine parts (a), (b), and (c) to conclude that m = n. e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.The amazing quadrilateral property-with coordinates Prove the quadrilateral property in Exercise 85, assuming the coordinates of P, Q, R, and S are P(x1, y1, 0), Q(x2, y2, 0), R(x3, y3, 0), and S (x4, y4, z4), where we assume P, Q, and R lie in the xy-plane without loss of generality.Sketch two nonzero vectors u and v with = 0. Sketch two nonzero vector u and v with = .Use Theorem 13.1 to computr the dot products i j, i k, and j k for the unit coordinate vector. What do you conclude about the angles between these vectors?Let u = 4i 3j. By inspection (not calculations), find the orthogonal projection of u onto i and onto j. Find the scalar component of u in the direction of i and in the direction of j.Express the dot product of u and v in terms of their magnitudes and the angle between them.Express the dot product of u and v in terms of the components of the vectors.Compute 2, 3, 6 1, 8, 3.4E5EFind the angle between u and v if scalvu = 2 and |u| = 4. Assume 0 .Find projvu if scalvu 2 and v 2,1,2.Use a dot product to determine whether the vectors u = 1,2,3 and v = 4,1,2 are orthogonal.9E10ESuppose v is a nonzero position vector in the xy-plane. How many position vectors with length 2 in the xy-plane are orthogonal to v?Suppose v is a nonzero position vector in xyz-space. How many position vectors with length 2 in xyz-space are orthogonal to v?13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28EAngles of a triangle For the given points P, Q, and R, find the approximate measurements of the angles of PQR. 71. P(0, 1, 3), Q(2, 2, 1), R(2, 2, 4)Angles of a triangle For the given points P, Q, and R, find the approximate measurements of the angles of PQR. 70. P(1, 4), Q(2, 7), R(2, 2)Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 25.Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 26.Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 27.Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 28.Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 29. u = 1, 4 and v = 4, 2Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 30. u = 10, 5 and v = 2, 6Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 33. u = 8, 0, 2 and v = 1, 3, 3Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 34. u = 5, 0, 15 and v = 0, 4, 239ECalculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 36. u = i + 4j + 7k and v = 2i 4j + 2k41EComputing work Calculate the work done in the following situations. 38. A stroller is pushed 20 m with a constant force of 10 N at an angle of 15 below the horizontal.43EComputing work Calculate the work done in the following situations. 40. A constant force F = 4, 3, 2 (in newtons) moves an object from (0, 0, 0) to (8, 6, 0). (Distance is measured in meters.)Computing work Calculate the work done in the following situations. 41. A constant force F = 40, 30 (in newtons) is used to move a sled horizontally 10 m.46EParallel and normal forces Find the components of the vertical force F = 0, 10 in the directions parallel to and normal to the following inclined planes. Show that the total force is the sum of the two component forces. 45. A plane that makes an angle of /3 with the positive x-axisParallel and normal forces Find the components of the vertical force F = 0, 10 in the directions parallel to and normal to the following inclined planes. Show that the total force is the sum of the two component forces. 46. A plane that makes an angle of =tan145 with the positive x-axis49EForces on an inclined plane An object on an inclined plane does not slide, provided the component of the objects weight parallel to the plane |Wpar| is less than or equal to the magnitude of the opposing frictional force |Fr|. The magnitude of the frictional force, in turn, is proportional to the component of the objects weight perpendicular to the plane |Wperp| (see figure). The constant of proportionality is the coefficient of static friction 0. Suppose a 100-lb block rests on a plane that is tilted at an angle of = 20 to the horizontal. a. Find |Wpar |and |Wperp | (Hint: it is not necessary to find Wpar and Wperp first.) b. The condition for the block not sliding is |Wpar| |Wperp|. If = 0.65, does the block slide? c. What is the critical angle above which the block slides?51EFor what value of a is the vector v = 4,3,7 orthogonal to w = a,8,3?For what value of c is the vector v = 2,5,c orthogonal to w = 3,2,9 ?Orthogonal vectors Let a and b be real numbers. 52. Find two vectors that are orthogonal to 0, 1, 1 and to each other.Orthogonal vectors Let a and b be real numbers. 49. Find all vectors 1, a, b orthogonal to 4, 8, 2.56E57EVectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of projvu. 54. Find another vector that has the same projection onto v = 1, 1 as u = 1, 2. Draw a picture.Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of projvu. 55. Let v = 1, 1. Give a description of the position vectors u such that projvu = projv 1, 2.Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of projvu. 56. Find another vector that has the same projection onto v = 1, 1, 1 as u = 1, 2, 3.Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of projvu. 57. Let v = 0, 0, 1. Give a description of all position vectors u such that projvu = projv 1, 2, 3.Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v. 58. u = 4, 3, v = 1, 1Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v. 59. u = 2, 2, v = 2, 1Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v. 60. u = 4, 3, 0, v = 1, 1, 1Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v. 61. u = 1, 2, 3, v = 2, 1, 1An alternative line definition Given a fixed point P0 (x0, y0) and a nonzero vector n = a,b the set of points Px,y for which P0P is orthogonal to n is a line (see figure). The vector n is called a normal vector or a vector normal to . 66. Show that the equation of the line passing through P0 (x0, y0) with a normal vector n = a,b is a(x x0) + b(y y0) = 0. (Hint: For a point P(x, y) on , examine n P0P.)An alternative line definition Given a fixed point P0 (x0, y0) and a nonzero vector n = a,b the set of points Px,y for which P0P is orthogonal to n is a line (see figure). The vector n is called a normal vector or a vector normal to . 67. Use the result of Exercise 66 to find an equation of the line passing through the point P0 (2, 6) with a normal vector n = (3, 7). Write the final answer in the form ax + by = c. 66. Show that the equation of the line passing through P0 (x0, y0) with a normal vector n = a,b is a(x x0) + b(y y0) = 0. (Hint: For a point P(x, y) on , examine n P0P.)68EAn alternative line definition Given a fixed point P0(x0, y0) and a nonzero vector n = a,b the set of points Px,y for which P0P is orthogonal to n is a line (see figure). The vector n is called a normal vector or a vector normal to . 69. Suppose a line is normal to n = 5,3. What is the slope of the line?Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 66. Show that I and J are orthogonal unit vectors.Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 67. Express I and J in terms of the usual unit coordinate vectors i and j. Then write i and j in terms of I and J.Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 68. Write the vector 2, 6 in terms of I and J.Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2,1/2, J=1/2,1/2,0, and K=1/2,1/2,1/2. a. Sketch I, J, and K and show that they are unit vectors. b. Show that I, J, and K are pairwise orthogonal. c. Express the vector 1, 0, 0 in terms of I, J, and K.Flow through a circle Suppose water flows in a thin sheet over the xy-plane with a uniform velocity given by the vector v = 1, 2; this means that at all points of the plane, the velocity of the water has components 1 m/s in the x-direction and 2 m/s in the y-direction (see figure). Let C be an imaginary unit circle (that does not interfere with the flow). a. Show that at the point (x, y) on the circle C, the outward-pointing unit vector normal to C is n = x, y. b. Show that at the point (cos , sin ) on the circle C, the outward-pointing unit vector normal to C is also n = cos , sin . c. Find all points on C at which the velocity is normal to C. d. Find all points on C at which the velocity is tangential to C. e. At each point on C, find the component of v normal to C. Express the answer as a function of (x, y) and as a function of . f. What is the net flow through the circle? That is, does water accumulate inside the circle?Heat flux Let D be a solid heat-conducting cube formed by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1. The heat flow at every point of D is given by the constant vector Q = 0, 2, 1. a. Through which faces of D does Q point into D? b. Through which faces of D does Q point out of D? c. On which faces of D is Q tangential to D (pointing neither in nor out of D)? d. Find the scalar component of Q normal to the face x = 0. e. Find the scalar component of Q normal to the face z = 1. f. Find the scalar component of Q normal to the face y = 0.Hexagonal circle packing The German mathematician Gauss proved that the densest way to pack circles with the same radius in the plane is to place the centers of the circles on a hexagonal grid (see figure). Some molecular structures use this packing or its three-dimensional analog. Assume all circles have a radius of 1 and let rij be the vector that extends from the center of circle i to the center of circle j, for i, j = 0, 1, , 6. a. Find r0j, for j = 1, 2, , 6. b. Find r12, r34, and r61. c. Imagine circle 7 is added to the arrangement as shown in the figure. Find r07, r17, r47, and r75.Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0), P(3,1,0), and Q(3,1,0). Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure). a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.) b. Let rij be the vector from the center of sphere I to the center of sphere J. Find rOP, rOQ, rPQ, rOR, and rPR.Properties of dot products Let u = u1, u2, u3, v = v1, v2, v3, and w = w1, w2, w3. Prove the following vector properties, where c is a scalar. 76. |u v| |u| |v|79E80E81EProperties of dot products Let u = u1, u2, u3, v = v1, v2, v3, and w = w1, w2, w3. Prove the following vector properties, where c is a scalar. 80. Distributive properties a. Show that (u + v) (u + v) = |u|2 + 2 u v + |v|2. b. Show that (u + v) (u + v) = |u|2 + |v|2 if u is orthogonal to v. c. Show that (u + v) (u v) = |u|2 |v|2.Direction angles and cosines Let v = a, b, c and let , , and be the angles between v and the positive x-axis, the positive y-axis, and the positive z-axis, respectively (see figure). a. Prove that cos2 + cos2 + cos2 = 1. b. Find a vector that makes a 45 angle with i and j. What angle does it make with k? c. Find a vector that makes a 60 angle with i and j. What angle does it make with k? d. Is there a vector that makes a 30 angle with i and j? Explain. e. Find a vector v such that = = . What is the angle?84E85ECauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because | cos | 1). This inequality, known as the CauchySchwarz Inequality, holds in any number of dimensions and has many consequences. 86. Geometric-arithmetic mean Use the vectors u=a,b and v=b,a to show that ab(a+b)/2, where a 0 and b 0.CauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because | cos | 1). This inequality, known as the CauchySchwarz Inequality, holds in any number of dimensions and has many consequences. 87. Triangle Inequality Consider the vectors u, v, and u + v (in any number of dimensions). Use the following steps to prove that |u + v| |u| + |v|. a. Show that |u + v|2 = (u + v) (u + v) = |u|2 + 2u v + |v|2. b. Use the CauchySchwarz Inequality to show that |u + v|2 (|u| + |v|)2. c. Conclude that |u + v| |u| + |v|. d. Interpret the Triangle Inequality geometrically in 2or 3.CauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because | cos | 1). This inequality, known as the CauchySchwarz Inequality, holds in any number of dimensions and has many consequences. 88. Algebra inequality Show that (u1+u2+u3)23(u12+u22+u32), for any real numbers u1, u2, and u3. (Hint: Use the CauchySchwarz Inequality in three dimensions with u = u1, u2, u3 and choose v in the right way.)Diagonals of a parallelogram Consider the parallelogram with adjacent sides u and v. a. Show that the diagonals of the parallelogram are u + v and u v. b. Prove that the diagonals have the same length if and only if u v = 0. c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.1QCExplain why the vector 2u 3v points in the same direction as the vector u v.A good check on a product calculation is to verify that u and v are orthogonal to the computed u v. In Example 4. Verify that u (u v) 0 and v (u v) 0. Example 4 Vector orthogonal to two vectors Find a vector orthogonal to the two vectors u = i + 6k and v = 2i 5j 3k.What is the magnitude of the cross product of two parallel vectors?2ESuppose u and v are nonzero vectors. What is the geometric relationship between u and v under each of the following conditions? a. UV = 0 b. U V= 0Use a geometric argument to explain why u (u v) = 0.Compute |u v| if u and v are unit vectors and the angle between them is /4.Compute |u v| if |u| = 3 and |v| = 4 and the angle between u and v is 2/3.7EFor any vector v in 3, explain why v v = 0.Explain how to use a determinant to compute u v.Explain how to find the torque produced by a force using cross products.Cross products from the definition Find the cross product u v in each figure. 7.Cross products from the definition Find the cross product u v in each figure. 8.Cross products from the definition Sketch the following vectors u and v. Then compute |u v| and show the cross product on your sketch. 9. u = 0, 2, 0, v = 0, 1, 014E15E16ECoordinate unit vectors Compute the following cross products. Then make a sketch showing the two vectors and their cross product. 15. j k18E19ECoordinate unit vectors Compute the following cross products. Then make a sketch showing the two vectors and their cross product. 18. 3j i21E22E23E24E25E26E27E28EArea of a parallelogram Find the area of the parallelogram that has two adjacent sides u and v. 21. u = 3i j, v = 3j + 2kArea of a parallelogram Find the area of the parallelogram that has two adjacent sides u and v. 22. u = 3i + 2k, v = i + j + kArea of a parallelogram Find the area of the parallelogram that has two adjacent sides u and v. 23. u = 2i j 2k, v = 3i + 2j kArea of a parallelogram Find the area of the parallelogram that has two adjacent sides u and v. 24. u = 8i + 2j 3k, v = 2i + 4j 4kArea of a triangle For the given points A, B, and C, find the area of the triangle with vertices A, B, and C. 25. A(0, 0, 0), B(3, 0, 1), C(1, 1, 0)Areas of triangles Find the area of the following triangles T. 57. The vertices of T are O(0, 0, 0), P(1, 2, 3), and Q(6, 5,4).Area of a triangle For the given points A, B, and C, find the area of the triangle with vertices A, B, and C. 27. A(5, 6, 2), B(7, 16, 4), C(6, 7, 3)Area of a triangle For the given points A, B, and C, find the area of the triangle with vertices A, B, and C. 28. A(1, 5, 3), B(3, 2, 1), C(0, 5, 1)Areas of triangles Find the area of the following triangles T. 55. The sides of T are u = 3, 3, 3,v = 6, 0, 6, and u v.Areas of triangles Find the area of the following triangles T. 54. The sides of T are u = 0, 6, 0, v = 4, 4, 4, and u v.Collinear points and cross products Explain why the points A, B, and C in 3 are collinear if and only if AB AC = 0.Collinear points Use cross products to determine whether the points A, B, and C are collinear. 50. A(3, 2, 1), B(5, 4, 7), and C(9, 8, 19)Collinear points Use cross products to determine whether the points A, B, and C are collinear. 51. A(3, 2, 1), B(1, 4, 7), and C(4, 10, 14)Orthogonal vectors Find a vector orthogonal to the given vectors. 36. 1, 2, 3and 2, 4, 1Orthogonal vectors Find a vector orthogonal to the given vectors. 35. 0, 1, 2and 2, 0, 3Orthogonal vectors Find a vector orthogonal to the given vectors. 37. 8, 0, 4and 8, 2, 1Computing torque Answer the following questions about torque. 41. Let r=OP=i+j+k. A force F = 20, 0, 0is applied at P. Find the torque about O that is produced.Computing torque Answer the following questions about torque. 42. Let r=OP=ij+2k. A force F = 10, 10, 0is applied at P. Find the torque about O that is produced.Computing torque Answer the following questions about torque. 43. Let r=OP=10i. Which is greater (in magnitude): the torque about O when a force F = 5i 5k is applied at P or the torque about O when a force F = 4i 3j is applied at P?Computing torque Answer the following questions about torque. 44. A pump handle has a pivot at (0, 0, 0) and extends to P(5, 0, 5). A force F = 1, 0, 10is applied at P. Find the magnitude and direction of the torque about the pivot.49E50E51EArm torque A horizontally outstretched arm supports a weight of 20 lb in a hand (see figure). If the distance from the shoulder to the elbow is 1 ft and the distance from the elbow to the hand is 1 ft, find the magnitude and describe the direction of the torque about (a) the shoulder and (b) the elbow. (The units of torque in this case are ft-lb.)Force on a moving charge Answer the following questions about force on a moving charge. 45. A particle with a positive unit charge (q = 1) enters a constant magnetic field B = i + j with a velocity v = 20k. Find the magnitude and direction of the force on the particle. Make a sketch of the magnetic field, the velocity, and the force.54E55EForce on a moving charge Answer the following questions about force on a moving charge. 48. A proton (q = 1.6 1019 C) with velocity 2 106 j m/s experiences a force in newtons of F = 5 1012 k as it passes through the origin. Find the magnitude and direction of the magnetic field at that instant.57EFinding an unknown Find the value of a such that a, a, 2 1, a, 3 = 2, 4, 2.59E60E61EExpress u, v, and w in terms of their components and show that u(v w) equals the determinant |u1u2u3v1v2v3w1w2w3|.63E64EScalar triple product Another operation with vectors is the scalar triple product, defined to be u (v w), for nonzero vectors u, v, and win 3. 65. Explain why the position vectors u, v, and w are coplanar if and only if |u (v w)| = 0. (Hint: See Exercise 63). 63. Consider the parallelepiped (slanted box) determined by the position vectors u, v, and w (see figure). Show that the volume of the parallelepiped is |u (v w) |, the absolute value of the scalar triple product.66E67EThree proofs Prove that u u = 0 in three ways. a. Use the definition of the cross product. b. Use the determinant formulation of the cross product. c. Use the property that u v = (v u).Associative property Prove in two ways that for scalars a and b, (au) (bv) = ab(u v). Use the definition of the cross product and the determinant formula.70E71E72EIdentities Prove the following identities. Assume that u, v, w, and x are nonzero vectors in 3. 73. u(v w) = (u w)v (u v)w Vector triple product74ECross product equations Suppose u and v are known nonzero vectors in 3. a. Prove that the equation u z = v has a nonzero solution z if and only if u v = 0. (Hint: Take the dot product of both sides with v.) b. Explain this result geometrically.Describe the line r = t k. for t . Describe the line r = t(i | j | 0k), for t .In the equation of the line x, y, zx0, y0, z0x1 x0, y1 y0, z1 z0 What value of t corresponds to the point P0(x0, y0, z0)? What value of t corresponds to the point P1(x1, y1, z1)?Find the distance between the point Q(1, 0, 3) and the line x, y, z = t 2,1,2. Note that P(0, 0, 0) lies on the line and v = 2,1,2is parallel to the line.Consider the equation of a plare in the form n P0P 0. Explain why the equation of the plane depends only on the direction. not on the length. of the normal vector n.Verify that in Example 6, the same equation for the plane results if either Q or R is used as the fixed point in the plane.Determine whether the planes 2x 3y + 6z = 12 and 6x + 8y + 2z = 1 are parallel, orthogonal, or neither.Find a position vector that is parallel to the line x = 2 + 4t, y = 5 8t, z = 9t.Find the parametric equations of the line r = 1,2,3 + t 4,0,6Explain how to find a vector in the direction of the line segment from P0(x0, y0, z0) to P1(x1, y1, z1).What is an equation of the line through the points P0(x0, y0, z0) and P1(x1, y1, z1)?Determine whether the plane x + y + z = 9 and the line x = t, y = t + 1, z = t + 2 are parallel, perpendicular, or neither. Be careful.
Determine whether the plane x + y + z = 9 and the line x = t, y = 2t + 1, z = t + 2 are parallel, perpendicular, or neither.Give two pieces of information which, taken together, uniquely determine a plane.Find a vector normal to the plane 2x 3y + 4z = 12.Where does the plane 2x 3y + 4z = 12 intersect the coordinate axes?Give an equation of the plane with a normal vector n = 1, 1, 1 that passes through the point (1, 0, 0).Equations of lines Find equations of the following lines. 9. The line through (0, 0, 1) in the direction of the vector v = 4, 7, 0Equations of lines Find equations of the following lines. 10. The Line through (3, 2, 1) in the direction of the vector v = 1, 2, 0Equations of lines Find equations of the following lines. 11. The line through (0, 0, 1) parallel to the y-axisEquations of lines Find both the parametric and the vector equations of the following lines. 14. The line through (2, 4, 3) parallel to the x-axisEquations of lines Find equations of the following lines. 13. The tine through (0, 0, 0) and (1, 2, 3)Equations of lines Find both the parametric and the vector equations of the following lines. 16. The line through (3, 4, 6) and (5, 1, 0)Equations of lines Find both the parametric and the vector equations of the following lines. 17. The line through (0, 0, 0) that is parallel to the line r=32t,5+8t,74t.Equations of lines Find equations of the following lines. 18. The line through (1, 3, 4) that is parallel to the line r(t) = 3 + 4t, 5 t, 7Equations of lines Find equations of the following lines. 19. The line through (0, 0, 0) that is perpendicular to both u = 1, 0, 2 and v = 0, 1, 1Equations of lines Find equations of the following lines. 20. The tine through (3, 4, 2) that is perpendicular to both u = 1, 1, 5 and v = 0, 4, 021EEquations of lines Find equations of the following lines. 22. The line through (0, 2, 1) that is perpendicular to both u = 4, 3, 5 and the z-axis23E24EEquations of lines Find both the parametric and the vector equations of the following lines. 25. The line that is perpendicular to the lines r = 4t,1+2t,3t and R = 1+s,7+2s,12+3s, and passes through the point of intersection of the lines r and REquations of lines Find both the parametric and the vector equations of the following lines. 26. The line that is perpendicular to the lines r = 2+3t,2t,3t and R = 6+s,8+2s,12+3s, and passes through the point of intersection of the lines r and RLine segments Find an equation of the line segment joining the first point to the second point. 25. (0, 0, 0) and (1, 2, 3)Line segments Find an equation of the line segment joining the first point to the second point. 26. (1, 0, 1) and (0, 2, 1)Line segments Find an equation of the line segment joining the first point to the second point. 27. (2, 4, 8) and (7, 5, 3)Line segments Find an equation of the line segment joining the first point to the second point. 28. (1, 8, 4) and (9, 5, 3)Parallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 31. r = 1,3,2+6,7,1 ; R = 10,6,14+s3,1,4Parallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 32. x = 2t, y = t + 2, z = 3t 1 and x = 5s 2, y = s + 4, z = 5s + 1Parallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 33. x = 4, y = 6 t, z = 1 + t and x = 3 7s, y = 1 + 4s, z = 4 sParallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 34. x = 4 + 5t, y = 2t, z = 1 + 3t and x = 10s, y = 6 + 4s, z = 4 + 6sParallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 35. x = 1 + 2t, y = 7 3t, z = 6 + t and x = 9 + 6t, y = 22 9t, z = 1 + 3tParallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 36. r=3,1,0+t4,6,4; R=0,5,4+s2,3,2Parallel, Intersecting, or skew lines Determine whether the following pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. 37. r=4+t,2t,1+3t; R=17t,6+14s,421sIntersecting lines and colliding particles Consider the lines r = 2+2t,8+t,10+3t and R = 6+s,102s,16s a. Determine whether the lines intersect (have a common point), and if so, find the coordinates of that point. b. If r and R describe the paths of two particles, do the particles collide? Assume t 0 and s 0 measure time in seconds, and that motion starts at s = t = 0.Distance from a point to a line Find the distance between the given point Q and the given line. 39. Q(5, 2, 9); x = 5t + 7, y = 2 t, z = 12t + 4Distance from a point to a line Find the distance between the given point Q and the given line. 40. Q(5, 6, 1); x= 1 + 3t, y = 3 4t, z = t + 1Billiards shot A cue ball in a billiards video game lies at P(25, 16) (see figure) Refer to Example 3, where we assume the diameter of each ball is 2.25 screen units, and pool balls are represented by the point at their center
The cue ball is aimed at an angle of 58° above the negative x-axis toward a target ball at A(5, 45). Do the balls collide?
b The cue ball is aimed at the point (50, 25) in an attempt to hit a target ball at B(76, 40). Do the balls collide?
The cue ball is aimed at an angle θ above the x-axis in the general direction of a target ball at C(75, 30). What range of angles (for 0 ≤ θ < π/2) will result in a collision? Express your answer in degrees.
42EEquations of planes Find an equation of the following planes. 43. The plane passing through the point P0(0,2,2) with a normal vector n=1,1,1Equations of planes Find an equation of the following planes. 44. The plane passing through the point P0(2,3,0) with a normal vector n = 1, 2, 3Equation of a plane Find an equation of the plane that is parallel to the vectors 1, 0 1 and 0, 2, 1 passing through the point (1, 2, 3).Equation of a plane Find an equation of the plane that is parallel to the vectors 1. 3, 1 and 4, 2, 0, passing through the point (3, 0, 2).Equations of planes Find an equation of the following planes. 47. The plane passing through the origin that is perpendicular to the line x t, y 1 +4t, z 7tEquations of planes Find an equation of the following planes. 48. The plane passing through the point (2, 3, 5) that is perpendicular to the line x (text not clear)2t, y (text not clear) 3t, z (text not clear) 5+4tEquations of planes Find an equation of the following planes. 17.The plane passing through the points (1, 0, 3), (0, 4, 2), and (1, 1, 1)Equations of planes Find an equation of the following planes. 19.The plane passing through the points (2, 1, 4), (1, 1, 1), and (4, 1, 1)Equations of planes Find an equation of the following planes. 51. The plane passing through the point P0(1,0,4) that is parallel to plane(text not clear)Equations of planes Find an equation of the following planes. 52. The plane passing through the point P0(0,2,2) that is parallel to the plane 2x + y z 1Equations of planes Find an equation of the following planes. 53. The plane containing the x-axis and the point P0(1,2,3)Equations of planes Find an equation of the following planes. 54. The plane containing the z-axis and the point P0(3,1,2)55E56EEquations of planes Find an equation of the following planes. 57. The plane passing through the point P0(4, 1, 2) and containing the line r = 2t2,2t,4t+158EParallel planes is the line x = t + 1, y = 2t + 3, z = 4t + 5 parallel to the plane 2x y = 2? If so, explain why and then find an equation of the plane containing the line that is parallel to the plane 2x y = 2Do the lines x = t, y = 2t + 1, z = 3t + 4 and x = 2s 2, y = 2s 1, z = 3s + 1 intersect each other at only one point? If so, find a plane that contains both lines.Properties of planes Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. 21.3x 2y + z = 662EProperties of planes Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. 23.x + 3y 5z 30 = 064EPairs of planes Determine whether the following pairs of planes are parallel, orthogonal, or neither. 25.x + y + 4z = 10 and x 3y + z = 10Pairs of planes Determine whether the following pairs of planes are parallel, orthogonal, or neither. 26.2x + 2y 3z = 10 and 10x 10y + 15z = 10Pairs of planes Determine whether the following pairs of planes are parallel, orthogonal, or neither. 27.3x + 2y 3z = 10 and 6x 10y + z = 10Pairs of planes Determine whether the following pairs of planes are parallel, orthogonal, or neither. 28.3x + 2y + 2z = 10 and 6x 10y + 19z = 10Equations of planes For the following sets of planes, determine which pairs of planes in the set are parallel, orthogonal, or identical. 29.Q: 3x 2y + z = 12; R: x + 2y/3 z/3 = 0; S: x + 2y + 7z = 1; T: 3x/2 y + z/2 = 6Equations of planes For the following sets of planes, determine which pairs of planes in the set are parallel, orthogonal, or identical. 30.Q: x + y z = 0; R: y + z = 0; S: x y = 0; T: x + y + z = 0Lines normal to planes Find an equation of the line passing through P0 and normal to the plane P. 73.P0(2, 1, 3); P: 2x 4y + z = 10Lines normal to planes Find an equation of the line passing through P0 and normal to the plane P. 74.P0(0, 10, 3); P: x + 4z = 2Intersecting planes Find an equation of the line of intersection of the planes Q and R. 35.Q: x + 2y + z = 1; R: x + y + z = 0Intersecting planes Find an equation of the line of intersection of the planes Q and R. 36.Q: x + 2y z = 1; R: x + y + z = 1Intersecting planes Find an equation of the line of intersection of the planes Q and R. 37.Q: 2x y + 3z 1 = 0; R: x + 3y + z 4 = 0Intersecting planes Find an equation of the line of intersection of the planes Q and R. 38.Q: x y 2z = 1; R: x + y + z = 1Line-plane intersections Find the point (if it exists) at which the following planes and lines intersect. 60. x = 3; r(t) = t, t, tLine-plane intersections Find the point (if it exists) at which the following planes and lines intersect. 62. y = 2; r(t) = 2t + 1,t + 4, t 6Line-plane intersections Find the point (if it exists) at which the following planes and lines intersect. 79. 3x + 2y 4z = 3 and x = 2t + 5, y = 3t 5, z = 4t 6Line-plane intersections Find the point (if it exists) at which the following planes and lines intersect. 80. 2x 3y + 3z = 2 and x = 3t, y = t, z = tExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The line r = 3, 1, 4+ t 6, 2, 8 passes through the origin. b. Any two nonparallel lines in 3 intersect. c. The plane x + y + z = 0 and the line x = t, y = t, z = t are parallel. d. The vector equations r = 1, 2, 3 + t 1, 1, 1 and R = 1, 2, 3) + t 2, 2, 2 describe the same line. e. The equations x + y z = 1 and x y + z = 1 describe the same plane. f. Any two distinct lines in 3 determine a unique plane. g. The vector 1, 5, 7 is perpendicular to both the line x = 1 + 5t, y = 3 t, z = 1 and the line x= 7t, y = 3, z = 3 + t.Distance from a point to a plane Suppose P is a point in the plane ax+ by+ cz = d. The distance from any point Q to the plane equals the length of the orthogonal projection of PQ onto a vector n = a, b, c normal to the plane. Use this information to show that the distance from Q to the plane is |PQn|/|n|.Find the distance from the point Q (6, 2, 4) to the plane 2x y + 2z = 4.Find the distance from the point Q (1, 2, 4) to the plane 2x 5z = 5.Symmetric equations for a line If we solve fort in the parametric equations of the line x = x0 + at, y = y0 + bt, z = z0 + ct, we obtain the symmetric equations xx0a=yy0b=zz0c, provided a, b, and c do not equal 0. 85. Find symmetric equations of the line r = 1, 2, 0 + t 4, 7, 2.Symmetric equations for a line If we solve fort in the parametric equations of the line x = x0 + at, y = y0 + bt, z = z0 + ct, we obtain the symmetric equations xx0a=yy0b=zz0c, provided a, b, and c do not equal 0. 86. Find parametric and symmetric equations of the line passing through the points P(1, 2, 3) and Q(2, 3, 1).Angle between planes The angle between two planes is the smallest angle between the normal vectors of the planes, where the directions of the normal vectors are chosen so that 0 /2. Find the angle between the planes 5x + 2y z = 0 and 3x + y + 2z = 0.88E89EOrthogonal plane Find an equation of the plane passing through (0, 2, 4) that is orthogonal to the planes 2x + 5y 3z = 0 and x + 5y + 2z = 8.Three intersecting planes Describe the set of all points (if any) at which all three planes x + 3z = 3, y + 4z = 6, and x + y + 6z = 9 intersect.Three intersecting planes Describe the set of all points (if any) at which all three planes x + 2y + 2z = 3, y + 4z = 6, and x + 2y + 8z = 9 intersect.To which coordinate axis in 3 is the cylinder z 2 ln x 0 parallel? To which coordinate axis in 3 is the cylinder y 4z2 1 parallel?Explain why the elliptic cylinder discussed in Example 1a, is a quadric surface. Example 1 Graphing cylinders Sketch the graphs of the following cylinders in 3. Identity the axis to which each cylinder is parallel. a. x2 + 4y2= 16 b. x sin z = 0Assume 0 c b a in the general equation of an ellipsoid. Along which coordinate axis does the ellipsoid have its longest axis? Its shortest axis?