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31-32 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 3

31-32 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 3

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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31

17. a = (1, -4, 1), b = (0, 2, -2) a=(-1,3, 4), b = (5, 2, 1) 19. a = 4i - 3j+ k, b = 2i - k =8i-j+4k, b = 4j + 2k 18. a 20. a = 21-22 Find, correct to the nearest degree, the three angles of the triangle with the given vertices. 21. P(2.0), Q(0, 3), R(3, 4) 22. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3) 23-24 Determine whether the given vectors are orthogonal, parallel, or neither. 23. (a) a = (9, 3), b = (-2, 6) (b) a = (4, 5, -2), b =(3,-1, 5) (c) a = -8i + 12j + 4k, b = 6i - 9j - 3k (d) a = 3i - j + 3k, b = 5i + 9j - 2k 24. (a) u = (-5, 4, -2), v = (3, 4, -1) (b) u = 9i - 6j + 3k, (c) u = (c, c, c), v = (c, 0, -c) v = -6i+4j - 2k 25. Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled. 26. Find the values of x such that the angle between the vectors (2, 1,-1), and (1, x, 0) is 45°. 27. Find a unit vector that is orthogonal to both i + j and i + k. 28. Find two unit vectors that make an angle of 60° with = (3,4). 30. x + 2y = 7, 29-30 Find the acute angle between the lines. 29. 2x - y = 3, 3x + y = 7 = 20 5x - y = 2 31-32 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 32. y sin x, y = cos x, 0≤x≤ π/2 33-37 Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 33. (2, 1, 2) 34. (6,3, -2) 35. i - 2j - 3k 36. i + j + k 37. (c, c, c), where c> 0 SECTION 12.3 The Dot Product 38. If a vector has direction angles a = third direction angle y. - 39-44 Find the scalar and vector projections of b onto a. 39. a = (-5, 12), b = (4, 6) 40. a = (1,4), b = (2,3) 41. a b = (3,-1, 1) = (4,7,-4), 42. a = (-1, 4, 8), b = (12, 1, 2) 43. a = 3i 3j+ k, 44. a = i + 2j + 3k, b = 5i - k TT/4 and 3 = π/3, find the b = 2i + 4j - k 45. Show that the vector orth, b = b - projab is orthogonal to a. (It is called an orthogonal projection of b.) 813 46. For the vectors in Exercise 40, find orth, b and illustrate by drawing the vectors a, b, proja b, and orth, b. 47. If a =(3, 0, -1), find a vector b such that compab = 2. 48. Suppose that a and b are nonzero vectors. (a) Under what circumstances is compab comp, a? (b) Under what circumstances is proja b = projь a? = 49. Find the work done by a force F = 8i - 6j +9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons. 50. 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 qed car 1 km? 02 51. A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of 40° above the horizontal moves the sled 80 ft. Find the work done by the force. 52. A boat sails south with the help of a wind blowing in the direc- tion S36°E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft. |ax₁ + by₁ +c| √a² + b² 53. Use a scalar projection to show that the distance from a point P₁(x₁, y₁) to the line ax + by + c = 0 is Use this formula to find the distance from the point (-2, 3) to the line 3x - 4y + 5 = 0. . 54. If r = (x, y, z), a = (a₁, a2, a3), and b = (b₁,b2, b3), show that the vector equation (ra) (r - b) = 0 represents a sphere, and find its center and radius. 55. Find the angle between a diagonal of a cube and one of its edges.
Transcribed Image Text:17. a = (1, -4, 1), b = (0, 2, -2) a=(-1,3, 4), b = (5, 2, 1) 19. a = 4i - 3j+ k, b = 2i - k =8i-j+4k, b = 4j + 2k 18. a 20. a = 21-22 Find, correct to the nearest degree, the three angles of the triangle with the given vertices. 21. P(2.0), Q(0, 3), R(3, 4) 22. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3) 23-24 Determine whether the given vectors are orthogonal, parallel, or neither. 23. (a) a = (9, 3), b = (-2, 6) (b) a = (4, 5, -2), b =(3,-1, 5) (c) a = -8i + 12j + 4k, b = 6i - 9j - 3k (d) a = 3i - j + 3k, b = 5i + 9j - 2k 24. (a) u = (-5, 4, -2), v = (3, 4, -1) (b) u = 9i - 6j + 3k, (c) u = (c, c, c), v = (c, 0, -c) v = -6i+4j - 2k 25. Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled. 26. Find the values of x such that the angle between the vectors (2, 1,-1), and (1, x, 0) is 45°. 27. Find a unit vector that is orthogonal to both i + j and i + k. 28. Find two unit vectors that make an angle of 60° with = (3,4). 30. x + 2y = 7, 29-30 Find the acute angle between the lines. 29. 2x - y = 3, 3x + y = 7 = 20 5x - y = 2 31-32 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 32. y sin x, y = cos x, 0≤x≤ π/2 33-37 Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 33. (2, 1, 2) 34. (6,3, -2) 35. i - 2j - 3k 36. i + j + k 37. (c, c, c), where c> 0 SECTION 12.3 The Dot Product 38. If a vector has direction angles a = third direction angle y. - 39-44 Find the scalar and vector projections of b onto a. 39. a = (-5, 12), b = (4, 6) 40. a = (1,4), b = (2,3) 41. a b = (3,-1, 1) = (4,7,-4), 42. a = (-1, 4, 8), b = (12, 1, 2) 43. a = 3i 3j+ k, 44. a = i + 2j + 3k, b = 5i - k TT/4 and 3 = π/3, find the b = 2i + 4j - k 45. Show that the vector orth, b = b - projab is orthogonal to a. (It is called an orthogonal projection of b.) 813 46. For the vectors in Exercise 40, find orth, b and illustrate by drawing the vectors a, b, proja b, and orth, b. 47. If a =(3, 0, -1), find a vector b such that compab = 2. 48. Suppose that a and b are nonzero vectors. (a) Under what circumstances is compab comp, a? (b) Under what circumstances is proja b = projь a? = 49. Find the work done by a force F = 8i - 6j +9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons. 50. 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 qed car 1 km? 02 51. A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of 40° above the horizontal moves the sled 80 ft. Find the work done by the force. 52. A boat sails south with the help of a wind blowing in the direc- tion S36°E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft. |ax₁ + by₁ +c| √a² + b² 53. Use a scalar projection to show that the distance from a point P₁(x₁, y₁) to the line ax + by + c = 0 is Use this formula to find the distance from the point (-2, 3) to the line 3x - 4y + 5 = 0. . 54. If r = (x, y, z), a = (a₁, a2, a3), and b = (b₁,b2, b3), show that the vector equation (ra) (r - b) = 0 represents a sphere, and find its center and radius. 55. Find the angle between a diagonal of a cube and one of its edges.
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Note: Since, the student has mentioned to solve question no. 31, so we are solving according to question no. 31.

To find the acute angle between the curves at their points of intersection. 

y=x2, y=x3

 

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