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Math Calculus Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

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 r ij be the vector from the center of sphere I to the center of sphere J. Find r OP , r OQ , r PQ , r OR , and r PR .

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 r ij be the vector from the center of sphere I to the center of sphere J. Find r OP , r OQ , r PQ , r OR , and r PR .

Solution Summary: The author explains that the coordinates of R are (2sqrt3,0) and

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

3rd Edition

ISBN: 9780136207764

Author: Briggs, William, Cochran, Lyle, Gillett, Bernard, SCHULZ, Eric

Publisher: PEARSON

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Chapter 13.3, Problem 77E

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).

Chapter 13.3, Problem 77E, Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0),

a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.)
b. Let r_ij be the vector from the center of sphere I to the center of sphere J. Find r_OP, r_OQ, r_PQ_, r_OR_, and r_PR.

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Chapter 13.3, Problem 76E

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Chapter 13 Solutions

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

Ch. 13.1 - Describe the length and direction of the vector 5v...Ch. 13.1 - Prob. 2QC Ch. 13.1 - Prob. 3QC Ch. 13.1 - Given the points P(2.3) and Q(4, 1), find the...Ch. 13.1 - Find vectors of length 10 parallel to the unit...Ch. 13.1 - Verify that the vector 513,1213 has length 1.Ch. 13.1 - Solve 3u | 4v = 12w for u.Ch. 13.1 - Interpret the following statement: Points have a...Ch. 13.1 - What is a position vector?Ch. 13.1 - Given a position vector v, why are there...

Ch. 13.1 - Use the points P(3.1) and Q(7.1) to find position...Ch. 13.1 - If u = u1, u2 and v = v1, v2, how do you find u +...Ch. 13.1 - Find two unit vectors parallel to 2,3.Ch. 13.1 - Is 1,1 a unit vector? Explain.Ch. 13.1 - Evaluate 3,1+2,4 and illustrate the sum...Ch. 13.1 - Prob. 9E Ch. 13.1 - Express the vector v = v1, v2 in terms of the unit...Ch. 13.1 - How do you compute |PQ| from the coordinates of...Ch. 13.1 - The velocity of a kayak on a lake is v=2,2,22....Ch. 13.1 - Vector operations Refer to the figure and carry...Ch. 13.1 - Vector operations Refer to the figure and carry...Ch. 13.1 - Vector operations Refer to the figure and carry...Ch. 13.1 - Vector operations Refer to the figure and carry...Ch. 13.1 - Prob. 17E Ch. 13.1 - Vector operations Refer to the figure and carry...Ch. 13.1 - Components and magnitudes Define the points O(0,...Ch. 13.1 - Prob. 20E Ch. 13.1 - Components and equality Define the points P(3, 1),...Ch. 13.1 - Components and equality Define the points P(3, 1),...Ch. 13.1 - Components and equality Define the points P(3, 1),...Ch. 13.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 13.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 13.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 13.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 13.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 13.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 13.1 - Prob. 30E Ch. 13.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 13.1 - Find a unit vector in the direction of v = 6,8.Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Find the vector v of length 6 that has the same...Ch. 13.1 - Find the vector v that has a magnitude of 10 and a...Ch. 13.1 - Designer vectors Find the following vectors. 73....Ch. 13.1 - Prob. 38E Ch. 13.1 - How do you find a vector of length 10 in the...Ch. 13.1 - Let v = 8,15. a. Find a vector in the direction of...Ch. 13.1 - Prob. 41E Ch. 13.1 - Prob. 42E Ch. 13.1 - Unit vectors Define the points P(4, 1), Q(3, 4),...Ch. 13.1 - Prob. 44E Ch. 13.1 - Prob. 45E Ch. 13.1 - Prob. 46E Ch. 13.1 - Unit vectors a. Find two unit vectors parallel to...Ch. 13.1 - Vectors from polar coordinates Suppose O is the...Ch. 13.1 - Vectors from polar coordinates Find the position...Ch. 13.1 - Prob. 50E Ch. 13.1 - Find the velocity v of an ocean freighter that is...Ch. 13.1 - Prob. 52E Ch. 13.1 - Airplanes and crosswinds Assume each plane flies...Ch. 13.1 - Prob. 54E Ch. 13.1 - Airplanes and crosswinds Assume each plane flies...Ch. 13.1 - A boat in a current The water in a river moves...Ch. 13.1 - Another boat in a current The water in a river...Ch. 13.1 - Prob. 58E Ch. 13.1 - Boat in a wind A sailboat floats in a current that...Ch. 13.1 - Prob. 60E Ch. 13.1 - Prob. 61E Ch. 13.1 - Prob. 62E Ch. 13.1 - Prob. 63E Ch. 13.1 - Prob. 64E Ch. 13.1 - Explain why or why not Determine whether the...Ch. 13.1 - Equal vectors For the points A(3, 4), B(6, 10),...Ch. 13.1 - Vector equations Use the properties of vectors to...Ch. 13.1 - Vector equations Use the properties of vectors to...Ch. 13.1 - Prob. 69E Ch. 13.1 - Solving vector equations Solve the following pairs...Ch. 13.1 - Prob. 71E Ch. 13.1 - Prob. 72E Ch. 13.1 - Prob. 73E Ch. 13.1 - Ant on a page An ant walks due east at a constant...Ch. 13.1 - Clock vectors Consider the 12 vectors that have...Ch. 13.1 - Three-way tug-of-war Three people located at A, B,...Ch. 13.1 - Additional Exercises 8185. Vector properties Prove...Ch. 13.1 - Additional Exercises 8185. Vector properties Prove...Ch. 13.1 - Vector properties Prove the following vector...Ch. 13.1 - Vector properties Prove the following vector...Ch. 13.1 - Vector properties Prove the following vector...Ch. 13.1 - Prob. 82E Ch. 13.1 - Magnitude of scalar multiple Prove that |cv| = |c|...Ch. 13.1 - Equality of vectors Assume PQ equals RS. Does it...Ch. 13.1 - Linear independence A pair of nonzero vectors in...Ch. 13.1 - Perpendicular vectors Show that two nonzero...Ch. 13.1 - Parallel and perpendicular vectors Let u = a, 5...Ch. 13.1 - The Triangle Inequality Suppose u and v are...Ch. 13.2 - Suppose the positive x-, y-, and z-axes point...Ch. 13.2 - To which coordinate planes are the planes x = 2...Ch. 13.2 - Describe the solution set of the equation (x 1)2...Ch. 13.2 - Which of the following vectors are parallel to...Ch. 13.2 - Which vector has the smaller magnitude: u = 3i j ...Ch. 13.2 - Explain how to plot the point (3, 2, 1) in 3.Ch. 13.2 - What is the y-coordinate of all points in the...Ch. 13.2 - Describe the plane x = 4.Ch. 13.2 - Prob. 4E Ch. 13.2 - Let u = 3, 5, 7 and v = 6, 5, 1. Evaluate u + v...Ch. 13.2 - What is the magnitude of a vector joining two...Ch. 13.2 - Which point is farther from the origin, (3, 1, 2)...Ch. 13.2 - Express the vector from P(1, 4, 6) to Q(1, 3, 6)...Ch. 13.2 - Points in 3 Find the coordinates of the vertices...Ch. 13.2 - Points in 3 Find the coordinates of the vertices...Ch. 13.2 - Points in 3 Find the coordinates of the vertices...Ch. 13.2 - Points in 3 Find the coordinates of the vertices...Ch. 13.2 - Plotting points in 3 For each point P(x, y, z)...Ch. 13.2 - Plotting points in 3 For each point P(x, y, z)...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Sketching planes Sketch the following planes in...Ch. 13.2 - Planes Sketch the plane parallel to the xy-plane...Ch. 13.2 - Prob. 22E Ch. 13.2 - Spheres and balls Find an equation or inequality...Ch. 13.2 - Spheres and balls Find an equation or inequality...Ch. 13.2 - Spheres and balls Find an equation or inequality...Ch. 13.2 - Spheres and balls Find an equation or inequality...Ch. 13.2 - Midpoints and spheres Find an equation of the...Ch. 13.2 - Midpoints and spheres Find an equation of the...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Prob. 34E Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Identifying sets Give a geometric description of...Ch. 13.2 - Prob. 39E Ch. 13.2 - Prob. 40E Ch. 13.2 - Prob. 41E Ch. 13.2 - Prob. 42E Ch. 13.2 - Prob. 43E Ch. 13.2 - Prob. 44E Ch. 13.2 - Unit vectors and magnitude Consider the following...Ch. 13.2 - Unit vectors and magnitude Consider the following...Ch. 13.2 - Unit vectors and magnitude Consider the following...Ch. 13.2 - Unit vectors and magnitude Consider the following...Ch. 13.2 - Prob. 49E Ch. 13.2 - Unit vectors and magnitude Consider the following...Ch. 13.2 - Flight in crosswinds A model airplane is flying...Ch. 13.2 - Another crosswind flight A model airplane is...Ch. 13.2 - Crosswinds A small plane is flying horizontally...Ch. 13.2 - Prob. 54E Ch. 13.2 - Prob. 55E Ch. 13.2 - Maintaining equilibrium An object is acted upon by...Ch. 13.2 - Explain why or why not Determine whether the...Ch. 13.2 - Sets of points Describe with a sketch the sets of...Ch. 13.2 - Sets of points Describe with a sketch the sets of...Ch. 13.2 - Sets of points Describe with a sketch the sets of...Ch. 13.2 - Sets of points 61. Give a geometric description of...Ch. 13.2 - Sets of points 62. Give a geometric description of...Ch. 13.2 - Sets of points 63. Give a geometric description of...Ch. 13.2 - Sets of points 64. Give a geometric description of...Ch. 13.2 - Prob. 65E Ch. 13.2 - Prob. 66E Ch. 13.2 - Write the vector v = 2, 4, 4 as a product of its...Ch. 13.2 - Find the vector of length 10 with the same...Ch. 13.2 - Find a vector of length 5 in the direction...Ch. 13.2 - Prob. 70E Ch. 13.2 - Prob. 71E Ch. 13.2 - Parallel vectors of varying lengths Find vectors...Ch. 13.2 - Parallel vectors of varying lengths Find vectors...Ch. 13.2 - Collinear points Determine the values of x and y...Ch. 13.2 - Collinear points Determine whether the points P,...Ch. 13.2 - Lengths of the diagonals of a box What is the...Ch. 13.2 - Three-cable load A 500-kg load hangs from three...Ch. 13.2 - Four-cable load A 500-lb load hangs from four...Ch. 13.2 - Possible parallelograms The points O(0, 0, 0),...Ch. 13.2 - Prob. 80E Ch. 13.2 - Midpoint formula Prove that the midpoint of the...Ch. 13.2 - Equation of a sphere For constants a, b, c, and d,...Ch. 13.2 - Prob. 83E Ch. 13.2 - Medians of a trianglewith coordinates In contrast...Ch. 13.2 - The amazing quadrilateral propertycoordinate free...Ch. 13.2 - The amazing quadrilateral property-with...Ch. 13.3 - Sketch two nonzero vectors u and v with = 0....Ch. 13.3 - Use Theorem 13.1 to computr the dot products i j,...Ch. 13.3 - Let u = 4i 3j. By inspection (not calculations),...Ch. 13.3 - Express the dot product of u and v in terms of...Ch. 13.3 - Express the dot product of u and v in terms of the...Ch. 13.3 - Compute 2, 3, 6 1, 8, 3.Ch. 13.3 - Prob. 4E Ch. 13.3 - Prob. 5E Ch. 13.3 - Find the angle between u and v if scalvu = 2 and...Ch. 13.3 - Find projvu if scalvu 2 and v 2,1,2.Ch. 13.3 - Use a dot product to determine whether the vectors...Ch. 13.3 - Prob. 9E Ch. 13.3 - Prob. 10E Ch. 13.3 - Suppose v is a nonzero position vector in the...Ch. 13.3 - Suppose v is a nonzero position vector in...Ch. 13.3 - Prob. 13E Ch. 13.3 - Prob. 14E Ch. 13.3 - Prob. 15E Ch. 13.3 - Prob. 16E Ch. 13.3 - Prob. 17E Ch. 13.3 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Prob. 21E Ch. 13.3 - Prob. 22E Ch. 13.3 - Prob. 23E Ch. 13.3 - Prob. 24E Ch. 13.3 - Prob. 25E Ch. 13.3 - Prob. 26E Ch. 13.3 - Prob. 27E Ch. 13.3 - Prob. 28E Ch. 13.3 - Angles of a triangle For the given points P, Q,...Ch. 13.3 - Angles of a triangle For the given points P, Q,...Ch. 13.3 - Sketching orthogonal projections Find projvu and...Ch. 13.3 - Sketching orthogonal projections Find projvu and...Ch. 13.3 - Sketching orthogonal projections Find projvu and...Ch. 13.3 - Sketching orthogonal projections Find projvu and...Ch. 13.3 - Calculating orthogonal projections For the given...Ch. 13.3 - Calculating orthogonal projections For the given...Ch. 13.3 - Calculating orthogonal projections For the given...Ch. 13.3 - Calculating orthogonal projections For the given...Ch. 13.3 - Prob. 39E Ch. 13.3 - Calculating orthogonal projections For the given...Ch. 13.3 - Prob. 41E Ch. 13.3 - Computing work Calculate the work done in the...Ch. 13.3 - Prob. 43E Ch. 13.3 - Computing work Calculate the work done in the...Ch. 13.3 - Computing work Calculate the work done in the...Ch. 13.3 - Prob. 46E Ch. 13.3 - Parallel and normal forces Find the components of...Ch. 13.3 - Parallel and normal forces Find the components of...Ch. 13.3 - Prob. 49E Ch. 13.3 - Forces on an inclined plane An object on an...Ch. 13.3 - Prob. 51E Ch. 13.3 - For what value of a is the vector v = 4,3,7...Ch. 13.3 - For what value of c is the vector v = 2,5,c...Ch. 13.3 - Orthogonal vectors Let a and b be real numbers....Ch. 13.3 - Orthogonal vectors Let a and b be real numbers....Ch. 13.3 - Prob. 56E Ch. 13.3 - Prob. 57E Ch. 13.3 - Vectors with equal projections Given a fixed...Ch. 13.3 - Vectors with equal projections Given a fixed...Ch. 13.3 - Vectors with equal projections Given a fixed...Ch. 13.3 - Vectors with equal projections Given a fixed...Ch. 13.3 - Decomposing vectors For the following vectors u...Ch. 13.3 - Decomposing vectors For the following vectors u...Ch. 13.3 - Decomposing vectors For the following vectors u...Ch. 13.3 - Decomposing vectors For the following vectors u...Ch. 13.3 - An alternative line definition Given a fixed point...Ch. 13.3 - An alternative line definition Given a fixed point...Ch. 13.3 - Prob. 68E Ch. 13.3 - An alternative line definition Given a fixed point...Ch. 13.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 13.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 13.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 13.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 13.3 - Flow through a circle Suppose water flows in a...Ch. 13.3 - Heat flux Let D be a solid heat-conducting cube...Ch. 13.3 - Hexagonal circle packing The German mathematician...Ch. 13.3 - Hexagonal sphere packing Imagine three unit...Ch. 13.3 - Properties of dot products Let u = u1, u2, u3, v =...Ch. 13.3 - Prob. 79E Ch. 13.3 - Prob. 80E Ch. 13.3 - Prob. 81E Ch. 13.3 - Properties of dot products Let u = u1, u2, u3, v =...Ch. 13.3 - Direction angles and cosines Let v = a, b, c and...Ch. 13.3 - Prob. 84E Ch. 13.3 - Prob. 85E Ch. 13.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 13.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 13.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 13.3 - Diagonals of a parallelogram Consider the...Ch. 13.4 - Prob. 1QC Ch. 13.4 - Explain why the vector 2u 3v points in the same...Ch. 13.4 - A good check on a product calculation is to verify...Ch. 13.4 - What is the magnitude of the cross product of two...Ch. 13.4 - Prob. 2E Ch. 13.4 - Suppose u and v are nonzero vectors. What is the...Ch. 13.4 - Use a geometric argument to explain why u (u v) =...Ch. 13.4 - Compute |u v| if u and v are unit vectors and the...Ch. 13.4 - Compute |u v| if |u| = 3 and |v| = 4 and the...Ch. 13.4 - Prob. 7E Ch. 13.4 - For any vector v in 3, explain why v v = 0.Ch. 13.4 - Explain how to use a determinant to compute u v.Ch. 13.4 - Explain how to find the torque produced by a force...Ch. 13.4 - Cross products from the definition Find the cross...Ch. 13.4 - Cross products from the definition Find the cross...Ch. 13.4 - Cross products from the definition Sketch the...Ch. 13.4 - Prob. 14E Ch. 13.4 - Prob. 15E Ch. 13.4 - Prob. 16E Ch. 13.4 - Coordinate unit vectors Compute the following...Ch. 13.4 - Prob. 18E Ch. 13.4 - Prob. 19E Ch. 13.4 - Coordinate unit vectors Compute the following...Ch. 13.4 - Prob. 21E Ch. 13.4 - Prob. 22E Ch. 13.4 - Prob. 23E Ch. 13.4 - Prob. 24E Ch. 13.4 - Prob. 25E Ch. 13.4 - Prob. 26E Ch. 13.4 - Prob. 27E Ch. 13.4 - Prob. 28E Ch. 13.4 - Area of a parallelogram Find the area of the...Ch. 13.4 - Area of a parallelogram Find the area of the...Ch. 13.4 - Area of a parallelogram Find the area of the...Ch. 13.4 - Area of a parallelogram Find the area of the...Ch. 13.4 - Area of a triangle For the given points A, B, and...Ch. 13.4 - Areas of triangles Find the area of the following...Ch. 13.4 - Area of a triangle For the given points A, B, and...Ch. 13.4 - Area of a triangle For the given points A, B, and...Ch. 13.4 - Areas of triangles Find the area of the following...Ch. 13.4 - Areas of triangles Find the area of the following...Ch. 13.4 - Collinear points and cross products Explain why...Ch. 13.4 - Collinear points Use cross products to determine...Ch. 13.4 - Collinear points Use cross products to determine...Ch. 13.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 13.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 13.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 13.4 - Computing torque Answer the following questions...Ch. 13.4 - Computing torque Answer the following questions...Ch. 13.4 - Computing torque Answer the following questions...Ch. 13.4 - Computing torque Answer the following questions...Ch. 13.4 - Prob. 49E Ch. 13.4 - Prob. 50E Ch. 13.4 - Prob. 51E Ch. 13.4 - Arm torque A horizontally outstretched arm...Ch. 13.4 - Force on a moving charge Answer the following...Ch. 13.4 - Prob. 54E Ch. 13.4 - Prob. 55E Ch. 13.4 - Force on a moving charge Answer the following...Ch. 13.4 - Prob. 57E Ch. 13.4 - Finding an unknown Find the value of a such that...Ch. 13.4 - Prob. 59E Ch. 13.4 - Prob. 60E Ch. 13.4 - Prob. 61E Ch. 13.4 - Express u, v, and w in terms of their components...Ch. 13.4 - Prob. 63E Ch. 13.4 - Prob. 64E Ch. 13.4 - Scalar triple product Another operation with...Ch. 13.4 - Prob. 66E Ch. 13.4 - Prob. 67E Ch. 13.4 - Three proofs Prove that u u = 0 in three ways. a....Ch. 13.4 - Associative property Prove in two ways that for...Ch. 13.4 - Prob. 70E Ch. 13.4 - Prob. 71E Ch. 13.4 - Prob. 72E Ch. 13.4 - Identities Prove the following identities. Assume...Ch. 13.4 - Prob. 74E Ch. 13.4 - Cross product equations Suppose u and v are known...Ch. 13.5 - Describe the line r = t k. for t . Describe the...Ch. 13.5 - In the equation of the line x, y, zx0, y0, z0x1 ...Ch. 13.5 - Find the distance between the point Q(1, 0, 3) and...Ch. 13.5 - Consider the equation of a plare in the form n P0P...Ch. 13.5 - Verify that in Example 6, the same equation for...Ch. 13.5 - Determine whether the planes 2x 3y + 6z = 12 and...Ch. 13.5 - Find a position vector that is parallel to the...Ch. 13.5 - Find the parametric equations of the line r =...Ch. 13.5 - Explain how to find a vector in the direction of...Ch. 13.5 - What is an equation of the line through the points...Ch. 13.5 - Determine whether the plane x + y + z = 9 and the...Ch. 13.5 - Determine whether the plane x + y + z = 9 and the...Ch. 13.5 - Give two pieces of information which, taken...Ch. 13.5 - Find a vector normal to the plane 2x 3y + 4z =...Ch. 13.5 - Where does the plane 2x 3y + 4z = 12 intersect...Ch. 13.5 - Give an equation of the plane with a normal vector...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find both the parametric and...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find both the parametric and...Ch. 13.5 - Equations of lines Find both the parametric and...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Prob. 21E Ch. 13.5 - Equations of lines Find equations of the following...Ch. 13.5 - Prob. 23E Ch. 13.5 - Prob. 24E Ch. 13.5 - Equations of lines Find both the parametric and...Ch. 13.5 - Equations of lines Find both the parametric and...Ch. 13.5 - Line segments Find an equation of the line segment...Ch. 13.5 - Line segments Find an equation of the line segment...Ch. 13.5 - Line segments Find an equation of the line segment...Ch. 13.5 - Line segments Find an equation of the line segment...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Parallel, Intersecting, or skew lines Determine...Ch. 13.5 - Intersecting lines and colliding particles...Ch. 13.5 - Distance from a point to a line Find the distance...Ch. 13.5 - Distance from a point to a line Find the distance...Ch. 13.5 - Billiards shot A cue ball in a billiards video...Ch. 13.5 - Prob. 42E Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equation of a plane Find an equation of the plane...Ch. 13.5 - Equation of a plane Find an equation of the plane...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Prob. 55E Ch. 13.5 - Prob. 56E Ch. 13.5 - Equations of planes Find an equation of the...Ch. 13.5 - Prob. 58E Ch. 13.5 - Parallel planes is the line x = t + 1, y = 2t + 3,...Ch. 13.5 - Do the lines x = t, y = 2t + 1, z = 3t + 4 and x =...Ch. 13.5 - Properties of planes Find the points at which the...Ch. 13.5 - Prob. 62E Ch. 13.5 - Properties of planes Find the points at which the...Ch. 13.5 - Prob. 64E Ch. 13.5 - Pairs of planes Determine whether the following...Ch. 13.5 - Pairs of planes Determine whether the following...Ch. 13.5 - Pairs of planes Determine whether the following...Ch. 13.5 - Pairs of planes Determine whether the following...Ch. 13.5 - Equations of planes For the following sets of...Ch. 13.5 - Equations of planes For the following sets of...Ch. 13.5 - Lines normal to planes Find an equation of the...Ch. 13.5 - Lines normal to planes Find an equation of the...Ch. 13.5 - Intersecting planes Find an equation of the line...Ch. 13.5 - Intersecting planes Find an equation of the line...Ch. 13.5 - Intersecting planes Find an equation of the line...Ch. 13.5 - Intersecting planes Find an equation of the line...Ch. 13.5 - Line-plane intersections Find the point (if it...Ch. 13.5 - Line-plane intersections Find the point (if it...Ch. 13.5 - Line-plane intersections Find the point (if it...Ch. 13.5 - Line-plane intersections Find the point (if it...Ch. 13.5 - Explain why or why not Determine whether the...Ch. 13.5 - Distance from a point to a plane Suppose P is a...Ch. 13.5 - Find the distance from the point Q (6, 2, 4) to...Ch. 13.5 - Find the distance from the point Q (1, 2, 4) to...Ch. 13.5 - Symmetric equations for a line If we solve fort in...Ch. 13.5 - Symmetric equations for a line If we solve fort in...Ch. 13.5 - Angle between planes The angle between two planes...Ch. 13.5 - Prob. 88E Ch. 13.5 - Prob. 89E Ch. 13.5 - Orthogonal plane Find an equation of the plane...Ch. 13.5 - Three intersecting planes Describe the set of all...Ch. 13.5 - Three intersecting planes Describe the set of all...Ch. 13.6 - To which coordinate axis in 3 is the cylinder z 2...Ch. 13.6 - Explain why the elliptic cylinder discussed in...Ch. 13.6 - Assume 0 c b a in the general equation of an...Ch. 13.6 - The elliptic paraboloid x=y23+z27 is a bowl-shaped...Ch. 13.6 - Which coordinate axis is the axis of the...Ch. 13.6 - Prob. 6QC Ch. 13.6 - To which coordinate axes are the following...Ch. 13.6 - Describe the graph of x = z2 in 3.Ch. 13.6 - What is a trace of a surface?Ch. 13.6 - What is the name of the surface defined by the...Ch. 13.6 - What is the name of the surface defined by the...Ch. 13.6 - What is the name of the surface defined by the...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Cylinders in 3 Consider the following cylinders in...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying quadric surfaces Identify the...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify the following...Ch. 13.6 - Identifying surfaces Identify the following...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Prob. 38E Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Prob. 42E Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Prob. 44E Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Prob. 47E Ch. 13.6 - Prob. 48E Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Quadric surfaces Consider the following equations...Ch. 13.6 - Prob. 52E Ch. 13.6 - Prob. 53E Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Identifying surfaces Identify and briefly describe...Ch. 13.6 - Prob. 59E Ch. 13.6 - Matching graphs with equations Match equations af...Ch. 13.6 - Explorations and Challenges 61. Solids of...Ch. 13.6 - Prob. 62E Ch. 13.6 - Prob. 63E Ch. 13.6 - Light cones The idea of a light cone appears in...Ch. 13.6 - Prob. 65E Ch. 13.6 - Hand tracking Researchers are developing hand...Ch. 13.6 - Designing a snow cone A surface, having the shape...Ch. 13.6 - Designing a glass The outer, lateral side of a...Ch. 13 - Explain why or why not Determine whether the...Ch. 13 - Prob. 2RE Ch. 13 - Prob. 3RE Ch. 13 - Prob. 4RE Ch. 13 - Prob. 5RE Ch. 13 - Working with vectors Let u = 2, 4, 5 and v = 6,...Ch. 13 - Working with vectors Let u = 2, 4, 5 and v = 6,...Ch. 13 - Prob. 8RE Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - working with vectors Let u = 2,4,5 , v = 6,10,2...Ch. 13 - Scalar multiples Find scalars a, b, and c such...Ch. 13 - Velocity vectors Assume the positive x-axis points...Ch. 13 - Prob. 18RE Ch. 13 - Spheres and balls Use set notation to describe the...Ch. 13 - Spheres and balls Use set notation to describe the...Ch. 13 - Spheres and balls Use set notation to describe the...Ch. 13 - Identifying sets. Give a geometric description of...Ch. 13 - Identifying sets. Give a geometric description of...Ch. 13 - Identifying sets. Give a geometric description of...Ch. 13 - Identifying sets. Give a geometric description of...Ch. 13 - Prob. 26RE Ch. 13 - Prob. 27RE Ch. 13 - Cross winds A small plane is flying north in calm...Ch. 13 - Prob. 29RE Ch. 13 - Canoe in a current A woman in a canoe paddles cue...Ch. 13 - Sets of points Describe the set of points...Ch. 13 - Angles and projections a. Find the angle between u...Ch. 13 - Prob. 33RE Ch. 13 - Prob. 34RE Ch. 13 - Computing work Calculate the work done in the...Ch. 13 - Computing work Calculate the work done in the...Ch. 13 - Prob. 37RE Ch. 13 - Inclined plane A 1804b map stands on a hillside...Ch. 13 - Area of a parallelogram Find the area of the...Ch. 13 - Area of a triangle Find the area of the triangle...Ch. 13 - Vectors normal to a plane Find a unit vector...Ch. 13 - Angle in two ways Find the angle between 2, 0, 2...Ch. 13 - Prob. 43RE Ch. 13 - Suppose you apply a force of |F| = 50 N near the...Ch. 13 - Prob. 45RE Ch. 13 - Lines in space Find an equation of the following...Ch. 13 - Lines in space Find an equation of the following...Ch. 13 - Lines in space Find an equation of the following...Ch. 13 - Lines in space Find an equation of the following...Ch. 13 - Lines in space Find an equation of the following...Ch. 13 - Equations of planes Consider the plane passing...Ch. 13 - Intersecting planes Find an equation of the line...Ch. 13 - Intersecting planes Find an equation of the line...Ch. 13 - Equations of planes Find an equation of the...Ch. 13 - Prob. 55RE Ch. 13 - Prob. 56RE Ch. 13 - Equations of planes Find an equation of the...Ch. 13 - Distance from a point to a line Find the distance...Ch. 13 - Distance from a point to a plane Find the distance...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Prob. 73RE Ch. 13 - Identifying surfaces Consider the surfaces defined...Ch. 13 - Prob. 75RE Ch. 13 - Designing a water bottle The lateral surface of a...

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