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All Textbook Solutions for Multivariable Calculus
Describe in words the region of 3 represented by the equation(s) or inequality. 34. x2 + y2 + z2 4Describe in words the region of 3 represented by the equation(s) or inequality. 35. 1 x2 + y2 + z2 536EDescribe in words the region of 3 represented by the equation(s) or inequality. 37. x2 + z2 938EWrite inequalities to describe the region. 39. The region between the yz-plane and the vertical plane x = 5Write inequalities to describe the region. 40. The solid cylinder that lies on or below the plane z = 8 and on or above the disk in the xy-plane with center the origin and radius 2Write inequalities to describe the region. 41. The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r RWrite inequalities to describe the region. 42. The solid upper hemisphere of the sphere of radius 2 centered at the origin43EConsider the points P such that the distance from P to A( 1, 5, 3) is twice the distance from P to B(6, 2, 2). Show that the set of all such points is a sphere, and find its center and radius.Find an equation of the set of all points equidistant from the points A(1, 5, 3) and B(6, 2, 2). Describe the set.Find the volume of the solid that lies inside both of the spheres x2 + y2 + z2 + 4x 2y + 4z + 5 = 0 and x2 + y2 + z2 = 447E48EAre the following quantities vectors or scalars? Explain. (a) The cost of a theater ticket (b) The current in a river (c) The initial flight path from Houston to Dallas (d) The population of tire worldWhat is the relationship between the point (4, 7) and the vector 4,7 ? Illustrate with a sketch.Name all the equal vectors in the parallelogram shown.4E5ECopy the vectors in the figure and use them to draw the following vectors. (a) a + b (b) a b (c) 12a (d) 3b (e) a + 2b (f) 2b aIn the figure, the tip of c and the tail of d are both the midpoint of QR. Express c and d in terms of a and b.If the vectors in the figure satisfy |u | = | v | = 1 and u + v + w = 0, what is | w |?Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 9. A(2, 1), B(1, 2)10EFind a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 11. A(3, 1), B(2, 3)Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 12. A(3, 2), B( 1, 0)Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 13. A(0, 3, 1), B(2, 3, 1)14E15EFind the sum of the given vectors and illustrate geometrically. 16. 3, 1, 1, 517E18E19E20EFind a + b, 4a + 2b, | a |, and | a b | 21. a = 4i 3j + 2k, b = 2i 4k22E23E24E25E26E27E28E29E30E31EFind the magnitude of the resultant force and the angle it makes with the positive x-axis. 33.33E34EA woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.36EA block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths 2 m and 3 m. The hoist weighs 350 N. The ropes, fastened at different heights, make angles of 50 and 38 with the horizontal. Find the tension in each rope and the magnitude of each tension.The tension T at each end of a chain has magnitude 25 N (see the figure). What is the weight of the chain?A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point The current in the canal flows at 3.5 km/h and the speed of his boat is 13 km/h. (a) In what direction should he steer? (b) How long will the trip take?Three forces act on an object. Two of the forces are at an angle of 100 to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has magnitude 4 N. Calculate the magnitude of the force that would exactly counterbalance these three forces.Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).42E43ELet C be the point on the line segment AB that is twice as far from B as it is from A. If a = OA, b = OB, and c = OC, show that c=23a+13b(a) Draw the vectors a = 3, 2, b = 2, 1, and c = 7, 1. (b) Show, by means of a sketch, that there are scalars s and t such that c = sa + tb. (c) Use the sketch to estimate the values of s and t. (d) Find the exact values of s and t.Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c = sa + tb for suitable scalars s and t. Then give an argument using components.If r = x, y, z and r0 = x0, y0, z0, describe the set of all points (x, y, z) such that | r r0 | = 1.48E49EProve Property 5 of vectors algebraically for the case n =3. Then use similar triangles to give a geometric proof.Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.52EWhich of the following expressions are meaningful? Which are meaningless? Explain. (a) (a b) c (b) (a b) c (c) | a | (b c) (d) a (b + c) (e) a b + c (f) | a | (b + c)2EFind a b. 3. a = 1.5, 0.4, b = 4, 6Find a b. 4. a = 6, 2, 3, b = 2, 5, 15EFind a b. 6. a. = p, p, 2p, b = 2q, q, q7E8EFind a b. 9. | a | = 7, | b | = 4, the angle between a and b is 30Find a b. 10. | a | = 80, | b | = 50, the angle between a and b is 3/4If u is a unit vector, find u v and u w.If u is a unit vector, find u v and u w.(a) Show that i j = j k = k i = 0. (b) Show that i i = j j = k k = 1.A street vendor sells a hamburgers, b hot dogs, and c soft drinks on a given day. He charges 4 for a hamburger. 2.50 for a hot dog, and 1 for a soft drink. If A = a, b, c and P = 4. 2.5, 1, what is the meaning of the dot product A P?Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) 15. a = 4, 3, b = 2. 116E17E18E19E20E21EFind, correct to the nearest degree, the three angles of the triangle with the given vertices. 22. A(1, 0, 1), B(3, 2, 0) C(1, 3, 3)23E24EUse vectors to decide whether the triangle with vertices P(l, 3, 2), Q(2, 0, 4), and R{6, 2, 5) is right-angled.Find the values of x such that the angle between the vectors 2, 1,1, and 1, x, 0 is 45.27EFind two unit vectors that make an angle of 60 with v = 3, 4.Find the acute angle between the lines. 29. 2x y = 3, 3x + y = 730E31EFind the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) 32. y = sin x, y cos x, 0 x /233EFind the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 34. 6, 3, 235E36EFind the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 37. c, c, c, where c 038E39E40EFind the scalar and vector projections of b onto a. 41. a = 4, 7, 4, b = 3, 1, 142E43E44E45E46E47ESuppose that a and b are nonzero vectors. (a) Under what circumstances is compa b = compb a? (b) Under what circumstances is proja b = prpjb a?49EA tow truck drags a stalled car along a road. The chain makes an angle of 30 with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km?51EA boat sails south with the help of a wind blowing in the direction S36E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.Use a scalar projection to show that the distance from a point P1(x1, y1) to the line ax + by + c = 0 is ax1+by1+ca2+b2 Use this formula to find the distance from the point (2, 3) to the line 3x 4y + 5 = 0.If r = x, y, z a = a1, a2, a3, and b = b1, b2, b3, show that the vector equation (r a) (r b) = 0 represents a sphere, and find its center and radius.55E56E57EIf c = | a | b + | b | a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.59ESuppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.Use Theorem 3 to prove the Cauchy-Schwarz Inequality: | a b| | a | | b |The Triangle Inequality for vectors is | a + b | | a | + | b | (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that | a + b |2 = (a + b) (a + b) and use Property 3 of the dot product.]The Parallelogram Law states that | a + b |2 + | a b |2 = 2 | a |2 + 2| b |2 (a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. (See the hint in Exercise 62.)64E65EFind the cross product a b and verify that it is orthogonal to both a and b. 1. a = 2, 3, 0, b = 1, 0, 52E3E4E5EFind the cross product a b and verify that it is orthogonal to both a and b. 6. a = ti + cos tj + sin tk, b = i sin tj + cos tk7EIf a = i 2k and b = j + k, find a b. Sketch a, b, and a b has vectors starting at the origin.Find the vector, not with determinants, but by using properties of cross products. 9. (i j) kFind the vector, not with determinants, but by using properties of cross products. 10. k (i 2j)Find the vector, not with determinants, but by using properties of cross products. 11. (j k) (k i)Find the vector, not with determinants, but by using properties of cross products. 12. (i + j) (i j)State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (a) a (b c) (b) a (b c) (c) a (b c) (d) a (b c) (e) (a b) (c d) (f) (a b) (c d)Find |u v| and determine whether u v is directed into the page or out of the page. 14.Find |u v| and determine whether u v is directed into the page or out of the page. 15.16EIf a = 2, 1, 3 and b = 4, 2, 1, find a b and b a.18EFind two unit vectors orthogonal to both 3, 2, 1 and 1, 1, 0.Find two unit vectors orthogonal to both j k and i + j.21E22E23EProve the property of cross products (Theorem 11). 24. Property 2: (ca) b = c(a b) = a (cb)25E26EFind the area of the parallelogram with vertices A(3, 0), B(1, 3). C(5, 2), and D(3. 1).Find the area of the parallelogram with vertices P(1, 0, 2), Q(3, 3, 3). R(7, 5, 8), and S(5,2, 7).(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the area of triangle PQR. 29. P(1, 0, 1), Q(2, 1, 3), R(4. 2, 5)(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the area of triangle PQR. 30. P(0, 0, 3), Q(4, 2, 0), R(3, 3, 1)(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the area of triangle PQR. 31. P(0, 2, 0), Q(4, 1, 2), R(5, 3, 1)32EFind the volume of the parallelepiped determined by the vectors a, b. and c. 33. a = 1, 2, 3), b = 1, 1, 2, c= 2, 1, 4Find the volume of the parallelepiped determined by the vectors a, b. and c. 34. a = i + j , b = j + k, c = i + j + kFind the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 35. P(2, 1, 0), Q(2. 3. 2), R(1, 4. 1), S(3, 6, 1)36E37E38E39E40E41ELet v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find the maximum and minimum values of the length of the vector u v. In what direction does u v point?43E44E(a) Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is d=aba where a = QR and b = QP. (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 6, 8) and R(1, 4, 7).47E48EProve that (a b) (a + b) = 2(a b).50EUse Exercise 50 to prove that a (b c) + b (c a) + c (a b) = 0Prove that (ab)(cd)=|acbcadbd|53EIf v1, v2, and v3 are noncoplanar vectors, let k1=v2v3v1(v2v3) k2=v3v1v1(v2v3) k3=v1v2v1(v2v3) (These vectors occur in the study of crystallography. Vectors of the form n1 v1 + n2 v2 + n3 v3, where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k3, form the reciprocal lattice.) (a) Show that ki is perpendicular to vj if i j. (b) Show that ki vi = 1 for i = 1, 2, 3. (c) Show that k1(k2k3)=1v1(v2v3)Determine whether each statement is true or false in 3. (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel.Find a vector equation and parametric equations for the line. 2. The line through the point (6, 5, 2) and parallel to the vector 1,3,23Find a vector equation and parametric equations for the line. 3. The line through the point (2, 2.4, 3.5) and parallel to the vector 3i + 2j k4EFind a vector equation and parametric equations for the line. 5. The line through the point (1, 0, 6) and perpendicular to the plane x + 3y + z = 56EFind parametric equations and symmetric equations for the line. 7. The line through the points (0, 12, 1) and (2, 1. 3)8EFind parametric equations and symmetric equations for the line. 9. The line through the points (8, 1, 4) and (3, 2, 4)Find parametric equations and symmetric equations for the line. 10. The line through (2, 1, 0) and perpendicular to both i + j and j + kFind parametric equations and symmetric equations for the line. 11. The line through (6, 2, 3) and parallel to the line 12x=13y=z+1Find parametric equations and symmetric equations for the line. 12. The line of intersection of the planes x + 2y + 3z = 1 and x y + z = 1Is the line through (4, 6, 1) and (2, 0, 3) parallel to the line through (10, 18,4) and (5, 3, 14)?14E(a) Find symmetric equations for the line that passes through the point (1, 5, 6) and is parallel to the vector 1.2.3. (b) Find the points in which the required line in part (a) intersects the coordinate planes.(a) Find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane. x y + 3z = 7. (b) In what points does this line intersect the coordinate planes?17E18EDetermine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. 19. L1: x = 3 + 2t, y = 4 t, z = 1+ 3t L2: x = 1 + 4s, y = 3 2s, z = 4 + 5s20E21E22EFind an equation of the plane. 23. The plane through the origin and perpendicular to the vector 1, 2, 524EFind an equation of the plane. 25. The plane through the point (1, 12, 3) and with normal vector i + 4j + k26EFind an equation of the plane. 27. The plane through the point (1, 1, 1) and parallel to the plane 5x y z = 6Find an equation of the plane. 28. The plane through the point (3, 2, 8) and parallel to the plane z = x + yFind an equation of the plane. 29. The plane through the point (1, 12, 13) and parallel to the plane x + y + z = 030EFind an equation of the plane. 31. The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)Find an equation of the plane. 32. The plane through the origin and the points (3, 2, 1) and (1, 1, 1)Find an equation of the plane. 33. The plane through the points (2, 1, 2), (3, 8, 6), and (2, 3, 1)34EFind an equation of the plane. 35. The plane that passes through the point (3, 5, 1) and contains the line x = 4 t, y = 2t 1, z = 3tFind an equation of the plane. 36. The plane that passes through the point (6, 1, 3) and contains the line with symmetric equations x/3 = y + 4 = z/2Find an equation of the plane. 37. The plane that passes through the point (3, 1, 4) and contains the line of intersection of the planes x + 2y + 3z = 1 and 2x y + z = 3Find an equation of the plane. 38. The plane that passes through the points (0, 2, 5) and (1, 3, 1) and is perpendicular to the plane 2z = 5x + 4y39E40EUse intercepts to help sketch the plane. 41. 2x + 5y + z = 1042E43E44E45E46EFind the point at which the line intersects the given plane. 47. 5x = y/2 = z + 2; 10x 7y + 3z+ 24 = 0Where does the line through (3, 1, 0) and (1, 5, 6) intersect the plane 2x + y z = 2?Find direction numbers for the line of intersection of the planes x + y + z = 1 and x + z = 0.50E51EDetermine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 9x 3y + 6z = 2, 2y = 6x + 4z53EDetermine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x y + 3z = 1, 3x + y z = 255E56E(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. x + y + z = 1, x + 2y + 2z = l(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. 3x 2y + z = 1, 2x + y 3z = 359EFind symmetric equations for the line of intersection of the planes. z = 2x y 5, z = 4x + 3y 561EFind an equation for the plane consisting of all points that equidistant from the points (2, 5, 5) and (6, 3, 1).63E(a) Find the point at which the given lines intersect: r = 1, 1,0 + t1,1,2 r = 2, 0, 2 + s1, 1, 0 (b) Find an equation of the plane that contains these lines.65E66E67E68E69E70E71E72EFind the distance between the given parallel planes. 2x 3y + z = 4, 4x 6y + 2z = 3Find the distance between the given parallel planes. 6z = 4y 2x, 9z = 1 3x + 6yShow that the distance between the parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is D=d1-d2a2+b2+c276EShow that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t, and x = 1 + 2s, y = 5 + 15s, z = 2 + 6s.Let L1 be the line through the origin and the point (2, 0, 1). Let L2 be the line through the points (1, 1, 1) and (41 1, 3). Find the distance between L1 and L2.Let L1 be the line through the points (1, 2, 6) and (2, 4, 8). Let L2 be the line of intersection of the planes P1, and P2, where P1 is the plane x y + 2z + 1 =0 and P2 is the plane through the points (3, 2, 1), (0, 0, 1), and (1, 2, 1). Calculate the distance between L1 and L2.81E82EIf a, b, and c are not all 0, show that the equation ax + by + cz + d = 0 represents a plane and a, b, c is a normal vector to the plane. Hint: Suppose a 0 and rewrite the equation in the form a(x+da)+b(y0)+c(z0)=0(a) What does the equation y = x2 represent as a curve in 2? (b) What does it represent as a surface in 3? (c) What does the equation z = y2 represent?(a)Sketch the graph of y = ex as a curve in 2. (b) Sketch the graph of y = ex as a surface in 3. (c) Describe and sketch the surface 2 = ey.Describe and sketch the surface. x2 + z2 = 1Describe and sketch the surface. 4x2 + y2 = 4Describe and sketch the surface. z = 1 y2Describe and sketch the surface. y = z27EDescribe and sketch the surface. z = sin y(a) Find and identify the traces of the quadric surface x2 + y2 z2 = 1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b) If we change the equation in part (a) to x2 y2 + z2 = 1, how is the graph affected? (c) What if we change the equation in part (a) to x2 + y2 + 2y z2 = 0?(a) Find and identify the traces of the quadric surface x2 y2 + z2 = 1 and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1. (b) If the equation in part (a) is changed to x2 y2 z2 = 1, what happens to the graph? Sketch the new graph.11EUse traces to sketch and identify the surface. 4x2 + 9y2 + 9z2 = 36Use traces to sketch and identify the surface. x2 = 4y2 + z214EUse traces to sketch and identify the surface. 9y2 + 4z2 = x2 + 3616E17EUse traces to sketch and identify the surface. 3x2 y2 + 3z2 = 019EUse traces to sketch and identify the surface. x = y2 z2Match the equation with its graph (labeled I-VIII). Give reasons for your choice. x2 + 4y2 + 9z2 = 1Match the equation with its graph (labeled I-VIII). Give reasons for your choice. 9x2 + 4y2 + z2 = 1Match the equation with its graph (labeled I-VIII). Give reasons for your choice. x2 y2 + z2 = 1Match the equation with its graph (labeled I-VIII). Give reasons for your choice. x2 + y2 z2 = 1Match the equation with its graph (labeled I-VIII). Give reasons for your choice. y = 2x2 + z2Match the equation with its graph (labeled I-VIII). Give reasons for your choice. y2 = x2 + 2z2Match the equation with its graph (labeled I-VIII). Give reasons for your choice. x2 + 2z2 = 1Match the equation with its graph (labeled I-VIII). Give reasons for your choice. y = x2 z229E30E31EReduce the equation to one of the standard forms, classify the surface, and sketch it. 4x2 y2 + 2z2 = 0