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All Textbook Solutions for Multivariable Calculus

Writing Vectors in Different Forms In Exercises 1 and 2, let u=PQ and v=PR and (a) write u and v in component form, (b) write u and v as the linear combination of the standard unit vectors i and j, (c) find the magnitudes of u and v, and (d) find 3u+v. P=(1,2),Q=(4,1),R=(5,4)2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE Finding the Angle Between Two Vectors In Exercises 23 and 24, find the angle between the vectors (a) in radians and (b) in degrees. u= 1,0,3 ,v= 2,2,1 25RE 26RE 27RE Finding the Projection of u onto v In Exercises 27 and 28, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. u=1,1,1,v=2,0,2 29RE 30RE 31RE 32RE Finding a Unit VectorFind a unit vector that is orthogonal to both u=2,10,8 and v=4,6,8.AreaFind the area of the parallelogram that has the vectors u=3,1,5 and v=2,4,1 as adjacent sides.35RE VolumeUse the triple scalar product to find the volume of the parallelepiped having adjacent edges u=2i+j, v=2j+k, and w=j+2k.Finding Parametric and Symmetric Equations In Exercises 37 and 38, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the two points. (For each line, write the direction numbers as integers.) (3,0,2),(9,11,6)38RE 39RE 40RE 41RE 42RE Finding an Equation of a Plane In Exercises 41-44, find an equation of the plane with the given characteristics. The plane contains the lines given by x12=y=z+1 and x+12=y1=z2 44RE 45RE 46RE Distance Find the distance between the planes 5x3y+z=2 and 5x3y+z=3.Distance Find the distance between the point (5,1,3) and the line given by x=1+t,y=32t, and z=5t.49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE Cylindrical-to-Rectangular ConversionIn Exercises 63 and 64, convert the point from cylindrical coordinates to rectangular coordinates. (2,3,3)65RE Spherical-to-Rectangular ConversionIn Exercises 65 and 66, convert the point from spherical coordinates to rectangular coordinates. (8,6,3)Converting a Rectangular EquationIn Exercises 67 and 68, convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. x2y2=2z 68RE Cylindrical-to-Rectangular Conversion In Exercises 69 and 70, find an equation in rectangular coordinates for the surface represented by the cylindrical equation, and sketch its graph. z=r2sin2+3rcos Cylindrical-to- Rectangular ConversionIn Exercises 69 and 70, find an equation in rectangular coordinates for the surface represented by the cylindrical equation, and sketch its graph. r=5z 71RE Spherical-to-Rectangular Conversion In Exercises 71 and 72, find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph. =9sec ProofUsing vectors, prove the Law of Sines: If a, b, and c are the three sides of the triangle shown in the figure, then sinAa=sinBb=sinCc.2PS 3PS Proof Using vectors, prove that the diagonals of a rhombus are perpendicular (see figure).Distance (a) Find the shortest distance between the point Q(2,0,0) and the line determined by the points P1(0,0,1) and P2(0,1,2). (b) Find the shortest distance between the point Q(2,0,0) and the line segment joining the points P1(0,0,1) and P2(0,1,2).6PS Volume (a) Find the volume of the solid bounded below by the paraboloid z=x2+y2 and above by the plane z=1. (b) Find the volume of the solid bounded below by the elliptic paraboloid z=x2a2+y2b2 and above by the plane z=k, where k0. (c) Show that the volume of the solid in part (b) is equal to one-half the product of the area of the base times the altitude, as shown in the figure.8PS 9PS 10PS 11PS 12PS 13PS 14PS 15PS 16PS Distance Between a Point and a PlaneConsider the plane that passes through the points P, R and S. Show that the distance from a point Q to this plane is Distance=|u(vw)|||uv|| where u=PR, v=PS, and w=PQ.18PS 19PS CONCEPT CHECK Scalar and Vector Describe the difference between a scalar and a vector. Give example of each.CONCEPT CHECK Vector Two points and a vector are given. Determine which point is the initial point and which point is theterminal point. Explain. P(2,1),Q(4,6), and v=6,7 Sketching a Vector In Exercises 3 and 4, (a) find the component form of the vector v and (b) sketch the vector with its initial point at the origin.4E Equivalent Vectors In Bunches 5-8, find the vector: u and v whose initial and terminal points are given. Show that u and v are equivalent. Initial PointTerminal Point u:(3,2)(5,6)v:(1,4)(3,8)Equivalent Vectors In Bunches 5-8, find the vector: u and v whose initial and terminal points are given. Show that u and v are equivalent. Initial PointTerminal Point u:(4,0)(1,8)v:(2,1)(7,7)Equivalent Vectors In Bunches 5-8, find the vector: u and v whose initial and terminal points are given. Show that u and v are equivalent. Initial PointTerminal Point u:(0,3)(6,2)v:(3,10)(9,5)Equivalent Vectors In Bunches 5-8, find the vector: u and v whose initial and terminal points are given. Show that u and v are equivalent. Initial PointTerminal Point u:(4,1)(11,4)v:(10,13)(25,10)Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form,(c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Terminal Initial Point (2,0) (5,5)Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form,(c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Terminal Initial Point (4,6) (3,6)Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form,(c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Terminal Initial Point (8,3) (6,1)Writing a Vector in Different Forms to Exercises 9-16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line cement. (b) Write the vector in component forth. (c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Initial PointTerminal Point (0,4)(5,1)13E Writing a Vector in Different Forms to Exercises 9-16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line cement. (b) Write the vector in component forth. (c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Initial PointTerminal Point (7,1)(3,1)Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form. (c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Initial Point Terminal Point (32,43) (12,3)Writing a Vector in Different Forms In Exercises 9-16. the initial and terminal points of a vector v are given, (a) Sketch the given directed line segment, (b) Write the vector in component form. (c) Write the vector as the linear combination of the standard unit vectors i and j. (d) Sketch the vector with its initial point at the origin. Initial Point Terminal Point (0.12,0.60)(0.84,1.25)Finding a Terminal Point In Exercise 17 and 18, the vector v and its initial point are given. Find the terminal point. v=1,3;Initialpoint:(4,2)Finding a Terminal Point In Exercise 17 and 18, the vector v and its initial point are given. Find the terminal point. v=4,9:Initialpoint:(5,3)19E 20E 21E Finding a Magnitude of a VectorIn Exercises 1924, find the magnitude of v. v=24,7 Finding a Magnitude of a VectorIn Exercises 1924, find the magnitude of v. v=i5j Finding a Magnitude of a VectorIn Exercises 1924, find the magnitude of v. v=3i+3j Sketching Scalar MultipliesIn Exercises 25 and 26, sketch each scalar multiple of v. v=3,5 (a) 2v (b) 3v (c) 72v (d) 23v Sketching Scalar MultipliesIn Exercises 25 and 26, sketch each scalar multiple of v. v=2,3 (a) 4v (b) 12v (c) 0v (d) 6v Using Vector Operations In Exercise 27 and 28, And (a) 23u , (b) 3v, (c) vu (d) 2u+5v. u=4,9,v=2,5 Using Vector Operations In Exercise 27 and 28, And (a) 23u , (b) 3v, (c) vu (d) 2u+5v. u=3,8,v=8,7 Sketching a Vector In Exercises 29-34, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to MathGrapht.com. u 30E Sketching a Vector In Exercises 29-34, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to MathGrapht.com. v Sketching a Vector In Exercises 29-34, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to MathGrapht.com. 12v Sketching a Vector In Exercises 29-34, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to MathGrapht.com. uv 34E Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has length 1. v=3,12 Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has length 1. v=5,15 Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has length 1. v=32,52 Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has length 1. v=6.2,3.4 Finding MagnitudesIn Exercises 3942, find the following. (a) u (b) v (c) u+v (d) uu (e) vv (f) u+vu+v u=1,1,v=1,2 40E Finding MagnitudesIn Exercises 3942, find the following. (a) u (b) v (c) u+v (d) uu (e) vv (f) u+vu+v u=1,12,v=2,3 42E Using the Triangle Inequality In Exercises 43 und 44, sketch a graph of u, v, and u+v. Then demonstrate the triangle inequality using the vectors u and v. u=2,1,v=5,4 44E 45E Finding a Vector In Exercises 45-48, find the vector v with the given magnitude and the same direction as u. Magnitude Direction v=4 u=1,1 47E 48E 49E 50E 51E 52E Finding a Vector In Exercises 53-56, find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. u=1,u=0v=3,v=45 Finding a Vector In Exercises 53-56, find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. u=4,u=0v=2,v=60 Finding a Vector In Exercises 53-56, find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. u=2,u=4v=1,v=2 Finding a Vector In Exercises 53-56, find the component form of u+v given the lengths of u and v and the angles that u and v make with the positive x-axis. u=5,u=0.5v=5,v=0.5 57E 58E 59E HOW DO YOU SEE IT? Use the figure to determine whether each statement is true or false. Justify your answer. (a) a=d (b) c=s (c) a+u=c (d) v+w=s (e) a+d=0 (f) uv=2(b+t)Finding Values In Exercises 61-66, And a and b such that v=au+bw, where u=1,2 and w=1,1. v=4,5 62E 63E 64E 65E 66E Finding Unit VectorsIn Exercises 6772, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. f(x)=x2,(3,9)68E 69E 70E Finding Unit Vectors In Exercises 67-72, find a unit vector (a) parallel to and (b) perpendicular to the graph of fat the given point. Then sketch the graph of f and sketch the vectors at the given point. f(x)=25x2,(3,4)72E 73E 74E 75E Numerical and Graphical Analysis Forces with magnitudes of 180 newtons and 275 newtons act on a hook (see figure). The angle between the two forces is degrees. (a) When =30, find the direction and magnitude of the resultant force. (b) Write the magnitude M and direction of the resultant force as functions of , where 0180. (c) Use a graphing utility to complete the table. 0 30 60 90 120 150 180 M (d) Use a graphing utility to graph the two functions M and . (e) Explain why one of the functions decreases for increasing values of , whereas the other does not.77E 78E Cable Tension In Exercises 79 and 80, determine the tension in the cable supporting the given load.Cable TensionIn Exercises 79 and 80, determine the tension in the cable supporting the given load.Projectile Motion A gun with a muzzle velocity of 1200 feet per second is fired at an angle of 6 above thehorizontal. Find the vertical and horizontal components of the velocity.82E Navigation A plane is flying with a bearing of 302. Its speed with respect to the air is 900 kilometers per hour. The wind at the planes altitude is from the southwest at100 kilometers per hour (see figure). What is the true direction of die plane, and what is its speed with respect to the ground?NavigationA plane flies at a constant groundspeed of 400 miles per hour due east and encounters a 50-mile-per-hour wind from the northwest. Find the airspeed and compass direction that will allow the plane to maintain its groundspeed and eastward direction.85E 86E 87E 88E 89E True or False? In Exercises 85-94, determine whether the statement is true or false. If it is false, explain why or give anexample that shows it is false. If u is a unit vector in the direction of v, then v=vu.91E True or False? In Exercises 8594, determine whether the statement is true or false. If it is false, explain why or give a example that shows it is false. If v=ai+bj=0, then a=b.93E 94E 95E 96E 97E Proof Prove that the vector w=uv+vu bisects the angle between n and v 99E PUTNAM EXAM CHALLENGE A coast artillery gun can fire at any angle of elevation between 0 and 90 in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant (=v0), determine the set H of points in the plane and above the horizontal which can be hit.CONCEPT CHECK Describing Coordinates A point in the three- dimensional coordinate system has coordinates (x0, y0, z0,) Describe what each coordinate measures.2E CONCEPT CHECK Comparing Graphs Describe the graph of x=4 on (a) the number line, (b) the two-dimensional coordinate system, and (c) the three-dimensional coordinate system.4E Plotting Points In Exercises 5-8. plot the points in the same three-dimensional coordinate system. (a) (2,1,3) (b) (1,2,1)6E 7E 8E Finding Coordinates of a Point In Exercises 9-12, find the coordinates of the point. The point is located three units behind the xz-plane. four units to the right of the xz-plane. and fist units above the xy-plane.Finding Coordinates of a PointIn Exercises 912, find the coordinates of the point. The point is located seven units in front of the yz-plane, two units to the left of the xz-plane, and one unit below the xy-plane.Finding Coordinates of a PointIn Exercises 912, find the coordinates of the point. The point is located on the x-axis, 12 units in front of the yz-plane.12E Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). z=1 14E Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). x=3 16E Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). y0 18E Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). y3 Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). x4 Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point (x, y, z) that satisfies the condition(s). xy0,z=3 22E Using the Three-Dimensional Coordinate System In Exercises 13-24, determine the location of a point ( x, y, z) that satisfies the condition(s). xyz0 24E 25E 26E 27E 28E Classifying a TriangleIn Exercises 2932, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (0,0,4),(2,6,7),(6,4,8)Classifying a TriangleIn Exercises 2932, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (3,4,1),(0,6,2),(3,5,6)Classifying a TriangleIn Exercises 2932, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (1,0,2),(1,5,2),(3,1,1)32E 33E 34E 35E 36E Finding the Equation of a Sphere In Exercises 37-42, find the standard equation of the sphere with the given characteristics. Center:(7,1,2);Radius;1 38E Finding the Equation of a SphereIn Exercises 3742, find the standard equation of the sphere with the given characteristics. Endpoints of a diameter: (2,1,3),(1,3,1)40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E Finding a Vector In Exercises 59-62, rind the vector z, given that u=1,2,3,v=2,2,1, and w=4,0,4. 2u+vw+3z=0 63E Parallel Vectors In Exercises 63-66, determine which of the vectors is/are parallel to z. Use a graphing utility to confirm your results. z=12i23j+34k (a) 6i4j+9k (b) i+43j32k (c) 12i+9k (d) 34ij+98k 65E Parallel Vectors In Exercises 63-66, determine which of the vectors is/are parallel to z. Use a graphing utility to confirm your results. z has initial point (5, 4, 1) and terminal point (2,4,4). (a) 7,6,2 (b) 14,16,6 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E Finding Unit Vectors In Exercises 79-82, find a unit vector (a) in the direction or r and (b) in the direction opposite of v. v=2,1,2 80E 81E 82E 83E 84E 85E 86E 87E Sketching a Vector In Exercises 87 und 88, sketch the vector v and write its component form. v lies in the xz-plane, has magnitude 5, and nukes an angle of 45 with die positive z-axis.89E 90E 91E 92E 93E 94E 95E 96E 97E Tower Guy Wire The guy wire supporting a 100-foot lower has a tension of 550 pounds. Using the distance shown in the figure, write the component form of the vector F representing the tension in the wire.Auditorium Lights The lights in an auditorium are 24-pound discs of radius 18 inches. Each disc is supported by three equally spaced cables that are L inches long (see figure). (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use a graphing utility and the function in part (a) to complete the table. L 20 25 50 35 40 45 50 T (c) Use a graphing utility to graph the function in pan (a). Determine the asymptotes of the graph. (d) Confirm the asymptotes of the graph in part (c) analytically. (e) Determine the minimum length of each cable when a cable is designed to carry a maximum load of 10 pounds.100E Load Supports Find the tension in each of the supporting cables in the figure when the weight of the crate is 500 new tons.102E 103E 1E Direction Cosines Consider the vector v=v1,v2,v3. What is the meaning of arccosv2v=30?3E 4E 5E 6E

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