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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. 49.

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. 49.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.1: Lines
Problem 2ECP
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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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