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All Textbook Solutions for Calculus: Early Transcendentals

7P (a) Prove a formula similar to the one in Problem 7 (a) but involving arccot instead of arctan (b) Find the sum of the series n=0arccot(n2+n+1).9P 10P Find the interval of convergence of n=1n3xn and find its sum.Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extends entirely beyond the table . In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. (Try it yourself with a deck of cards.) Consider centers of mass. FIGURE FOR PROBLEM 12 Find the sum of the series n=2ln(11n2).14P 15P 16P 17P 18P 19P 20P 21P 22P 23P (a) Show that the Maclaurin series of the function f(x)=x1xx2isn=1fnxn wherefn is the nth Fibonacci number, that is, f1 = 1, f2 = 1, and fn = fn1 + fn2 for n 3. [Hint: Write x/(1 x x2) = c0 + c1x + c2x2 + and multiply both sides of this equation by 1 x x2.] (b) By writing f(x) as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way. find an explicit formula for the nth Fibonacci number.Let u=1+x33!+x66!+x99!+v=x+x44!+x77!+x1010!+w=x22!+x55!+x88!+ Show that u3 + v3 + w3 3uvw = 1.26P Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?Sketch the points (1, 5, 3), (0, 2, 3), (3, 0, 2), and (2, 2, 1) on a single set of coordinate axes.Which of the points A(4, 0, 1), B(3, 1, 5), and C(2, 4, 6) is closest to the yz-plane? Which point lies in the xz-plane?4E What does the equation x = 4 represent in 2? What does it represent in 3? Illustrate with sketches.What does the equation y = 3 represent in 3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points (x, y, z) such that y = 3 and z = 5. Illustrate with a sketch.Describe and sketch the surface in 3 represented by the equation x + y = 2.Describe and sketch the surface in 3 represented by the equation x2 + z2 = 9.Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? 9. P(3. 2, 3), Q(7.0. 1), R(1, 2, 1)Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? 10. P(2, 1, 0), Q(4, 1, 1), R(4, 5, 4)Determine whether the points lie on a straight line. (a) A(2, 4, 2), B(3, 7, 2), C(1, 3, 3) (b) D(0,5, 5), E(1, 2, 4), F(3, 4, 2)Find the distance from (4, 2, 6) to each of the following. (a) The xy-plane (b) The yz-plane (c) The xz-plane (d) The x-axis (e) The y-axis (f) The z-axis Find an equation of the sphere with center (3, 2, 5) and radius 4. What is the intersection of this sphere with the yz-plane?Find an equation of the sphere with center (2, 6, 4) and radius 5. Describe its intersection with each of the coordinate planes.Find an equation of the sphere that passes through the point (4, 3, 1) and has center (3, 8, 1).Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).Show that the equation represents a sphere, and find its center and radius. 17. x2 + y2 + z2 2x - 4y + 8z = 15 18E 19E 20E (a) Prove that the midpoint of the line segment from P1(x1, y1, z1) to P2(x2, y2, z2) is (x1+x22,y1+y22,z1+z22) (b) Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(2, 0, 5), and C(4, 1, 5).(A median of a triangle is a line segment that joins a vertexto the midpoint of the opposite side.)Find an equation of a sphere if one of its diameters has end points (5, 4, 3) and (1, 6, 9).Find equations of the spheres with center (2, 3, 6) that touch (a) the xy-plane, (b) the yz-plane, (c) the xz-plane.Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant.Describe in words the region of 3 represented by the equation(s) or inequality. 25. x = 5 Describe in words the region of 3 represented by the equation(s) or inequality. 26. y = 2 Describe in words the region of 3 represented by the equation(s) or inequality. 27. y 8 Describe in words the region of 3 represented by the equation(s) or inequality. 28. z 1 Describe in words the region of 3 represented by the equation(s) or inequality. 29. 0 z 6 Describe in words the region of 3 represented by the equation(s) or inequality. 30. y2 = 4 Describe in words the region of 3 represented by the equation(s) or inequality. 31. x2 + y2 = 4, z = 1 Describe in words the region of 3 represented by the equation(s) or inequality. 32. x2 + y2 = 4 Describe in words the region of 3 represented by the equation(s) or inequality. 33. x2 + y2 + z2 = 4 Describe in words the region of 3 represented by the equation(s) or inequality. 34. x2 + y2 + z2 4 Describe in words the region of 3 represented by the equation(s) or inequality. 35. 1 x2 + y2 + z2 5 Describe in words the region of 3 represented by the equation(s) or inequality. 36. x = z Describe in words the region of 3 represented by the equation(s) or inequality. 37. x2 + z2 9 Describe in words the region of 3 represented by the equation(s) or inequality. 38. x2 + y2 + z2 2z Write inequalities to describe the region. 39. The region between the yz-plane and the vertical plane x = 5 Write 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 2 Write 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 R Write inequalities to describe the region. 42. The solid upper hemisphere of the sphere of radius 2 centered at the origin The figure shows a line L1 in space and a second line L2, which is the projection of L1 onto the xy-plane. (In other words, the points on L2 are directly beneath, or above, the points on L1.) (a) Find the coordinates of the point P on the line L1. (b) Locate on the diagram the points A, B, and C, where the line L1 intersects the xy-plane, the yz-plane, and the xz-plane, respectively.Consider 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 = 4 Find the distance between the spheres x2 + y2 + z2 = 4 and x2 + y2 + z2 = 4x + 4y + 4z 11.Describe and sketch a solid with the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle.Are 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 world What 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.4E 5E Copy 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 a In 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)Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 10. 4(5, 1), B(3, 3)Find 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)Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 14. A(0, 6, 1), B(3, 4, 4)Find the sum of the given vectors and illustrate geometrically. 15. 1, 4, 6, 2 Find the sum of the given vectors and illustrate geometrically. 16. 3, 1, 1, 5 Find the sum of the given vectors and illustrate geometrically. 17. 3, 0, 1, 0, 8, 0 Find the sum of the given vectors and illustrate geometrically. 18. 1, 3, 2, 0, 0, 6 Find a + b, 4a + 2b, | a |, and | a b | 19. a = 3, 4, b = 9, 1 Find a + b, 4a + 2b, | a |, and | a b | 20. a = 5i + 3j, b = i 2j Find a + b, 4a + 2b, | a |, and | a b | 21. a = 4i 3j + 2k, b = 2i 4k Find a + b, 4a + 2b, | a |, and | a b | 22. a = 8, 1, 4, b = 5, 2, 1 Find a unit vector that has the same direction as the given vector. 23. 6, 2 24E Find a unit vector that has the same direction as the given vector. 25. 8i j + 4k Find the vector that has the same direction as 6, 2, 3 but has length 4.27E 28E If v lies in the first quadrant and makes an angle /3 with the positive x-axis and | v | = 4, find v in component form.30E 31E Find the magnitude of the resultant force and the angle it makes with the positive x-axis. 33.Find the magnitude of the resultant force and the angle it makes with the positive x-axis. 33.34E A 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.36E A 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).42E If A, B, and C are the vertices of a triangle, find AB+BC+CA Let 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.If r = x, y, r1 = x1, y1, and r2 = x2, y2, describe the set of all points (x, y) such that | r r1 | + | r r2| = k, where k | r1 r2|.Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case n = 2.Prove 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.52E Which 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)Find a b. 2. a = 5, 2, b = 3, 4 Find a b. 3. a = 1.5, 0.4, b = 4, 6 Find a b. 4. a = 6, 2, 3, b = 2, 5, 1 Find a b. 5. a = 4, 1, 14, b = 6, 3, 8 Find a b. 6. a. = p, p, 2p, b = 2q, q, q Find a b. 7. a = 2i + j, b = i j + k Find a b. 8. a = 3i + 2j k, b = 4i + 5k Find a b. 9. | a | = 7, | b | = 4, the angle between a and b is 30 Find a b. 10. | a | = 80, | b | = 50, the angle between a and b is 3/4 If 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. 1 16E 17E 18E 19E Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) 20. a = 8i j + 4k, b = 4j + 2k Find, correct to the nearest degree, the three angles of the triangle with the given vertices. 21. P(2,0), Q(2(0,3), R(3.4)Find, 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)23E 24E Use 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.Find a unit vector that is orthogonal to both i + j and i + k.Find 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 = 7 30E Find 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.) 31. y = x2, y = x3 Find 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 /2 Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 33. 2, 1, 2 34E 35E 36E Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 37. c, c, c, where c 0 38E Find the scalar and vector projections of b onto a. 39. a = 5, 12, b = 4, 6 Find the scalar and vector projections of b onto a. 40. a = 1, 4, b = 2, 3 Find the scalar and vector projections of b onto a. 41. a = 4, 7, 4, b = 3, 1, 1 Find the scalar and vector projections of b onto a. 42. a = 1, 4, 8, b = 12, 1, 2 43E 44E 45E 46E 47E Suppose 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?49E A 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?51E A 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.55E 56E 57E If c = | a | b + | b | a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.Prove Properties 2.4, and 5 of the dot product (Theorem 2).Suppose 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 |62E 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.)64E 65E Find the cross product a b and verify that it is orthogonal to both a and b. 1. a = 2, 3, 0, b = 1, 0, 5 2E 3E 4E 5E Find 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 tk 7E If 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) k Find 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. 15.16E If a = 2, 1, 3 and b = 4, 2, 1, find a b and b a.18E Find 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.Show that 0 a = 0 = a 0 for any vector a in V3.22E 23E 24E 25E 26E Find 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)(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the area of triangle PQR. 32. P(2, 3, 4) Q(1, 2, 2), R(3, 1, 3)Find 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, 4 Find the volume of the parallelepiped determined by the vectors a, b. and c. 34. a = i + j , b = j + k, c = i + j + k Find 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)Find the volume of the parallelepiped with adjacent edges PQ, PR. and PS. 36. P(3, 0, 1), Q(1, 2, 5), R(5, 1, 1), S(0, 4, 2)Use the scalar triple product to verify that the vectors u = i + 5j 2k, v = 3i j, and w = 5i + 9j 4k are coplanar.38E 39E 40E 41E Let 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?If a b = 3 and a b = 1, 2, 2, find the angle between a and b.(a) Find all vectors v such that 1, 2, 1 x v = 3, 1, 5) (b) Explain why there is no vector v such that 1, 2, 1 v = 3, 1, 5 (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).Show that | a b |2 = | a |2| b |2 (a b)2 48E Prove that (a b) (a + b) = 2(a b).Prove Properly 6 of cross products, that is, a (b c) = (a c)b (a b)c Use Exercise 50 to prove that a (b c) + b (c a) + c (a b) = 0 Prove that (ab)(cd)=|acbcadbd|Suppose that a 0. (a) If a b = a c, does it follow that b = c? (b) If a b = a c, does it follow that b = c? (c) If a b = a c and a b = a c, does it follow that b = c?If 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 1,3,23 Find 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 k Find a vector equation and parametric equations for the line. 4. The line through the point (0, 14, 10) and parallel to the line x = 1 + 2t, y = 6 3t, z = 3 + 9t Find 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 = 5 Find parametric equations and symmetric equations for the line. 6. The line through the origin and the point (4, 3, 1)Find parametric equations and symmetric equations for the line. 7. The line through the points (0, 12, 1) and (2, 1. 3)Find parametric equations and symmetric equations for the line. 8. The line through the points (1, 2.4, 4.6) and (2.6, 1.2, 0.3)Find 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 + k Find parametric equations and symmetric equations for the line. 11. The line through (6, 2, 3) and parallel to the line 12x=13y=z+1 Find parametric equations and symmetric equations for the line. 12. The line of intersection of the planes x + 2y + 3z = 1 and x y + z = 1 Is the line through (4, 6, 1) and (2, 0, 3) parallel to the line through (10, 18,4) and (5, 3, 14)?Is the line through (2, 4, 0) and (1, 1, 1) perpendicular to the line through (2, 3, 4) and (3, 1, 8)?(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?Find a vector equation for the line segment from (6, 1, 9) to (7, 6, 0).Find parametric equations for the line segment from (2, 18. 31) to (11. 4, 48).Determine 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 + 5s Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. 20. L1: x = 5 12t, y = 3 + 9t, z = 1 3t L2: x = 3 + 8s, y = 6s, z = 7 + 2s Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. 21. L1:x21=y32=z13L2:x31=y+43=z27 Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. 22.L1:x1=y11=z23L2:x22=y42=z7 Find an equation of the plane. 23. The plane through the origin and perpendicular to the vector 1, 2, 5 24E 25E Find an equation of the plane. 26. The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 t, z = 3 + 4t Find an equation of the plane. 27. The plane through the point (1, 1, 1) and parallel to the plane 5x y z = 6 Find an equation of the plane. 28. The plane through the point (3, 2, 8) and parallel to the plane z = x + y Find an equation of the plane. 29. The plane through the point (1, 12, 13) and parallel to the plane x + y + z = 0 Find an equation of the plane. 30. The plane that contains the line x = 1 + t, y = 2 t, z = 4 3t and is parallel to the plane 5x + 2y + z = 1 Find 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)Find an equation of the plane. 34. The plane through the points (3, 0, 1), (2, 2, 3), and (7, 1, 4)Find 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 = 3t Find 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/2 Find 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 = 3 Find 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 + 4y Find an equation of the plane. 39. The plane that passes through the point (1, 5, 1) and is perpendicular to the planes 2x + y 2z = 2 and x + 3z = 4 Find an equation of the plane. 40. The plane that passes through the line of intersection of the planes x z = 1 and y + 2z = 3 and is perpendicular to the plane x + y 2z = 1 Use intercepts to help sketch the plane. 41. 2x + 5y + z = 10 Use intercepts to help sketch the plane. 42. 3x + y + 2z = 6 Use intercepts to help sketch the plane. 43. 6x 3y + 4z = 6 Use intercepts to help sketch the plane. 44. 6x + 5y 3z = 15 45E 46E Find the point at which the line intersects the given plane. 47. 5x = y/2 = z + 2; 10x 7y + 3z+ 24 = 0 Where 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.Find the cosine of the angle between the planes x + y + z = 0 and x + 2y + 3z = 1.Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 4y 3z = 1, 3x + 6y + 7z = 0 Determine 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 + 4z Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 2y z = 2, 2x 2y + z = 1 54E Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 2x 3y = z, 4x = 3 + 6y + 2z Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 5x + 2y + 3z = 2, y = 4x 6z (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 = 3 Find symmetric equations for the line of intersection of the planes. 5x 2y 2z = 1, 4x + y + z = 6 Find symmetric equations for the line of intersection of the planes. z = 2x y 5, z = 4x + 3y 5 Find an equation for the plane consisting of all points that are equidistant from the points (1, 0, 2) and (3, 4, 0).Find an equation for the plane consisting of all points that equidistant from the points (2, 5, 5) and (6, 3, 1).

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