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8-4 Vectors in Three-Dimensional Space
Find the component form and magnitude of
unit vector in the direction of
.
33. A(−5, 12, 17), B(6, −11, 4)
with the given initial and terminal points. Then find a
SOLUTION: Find the component form of
.
Use the component form to find the magnitude of
.
Using this magnitude and component form, find a unit vector u in the direction of
.
35. TETHERBALL In the game of tetherball, a ball is attached to a 10-foot pole by a length of rope. Two players hit
the ball in opposing directions in attempt to wind the entire length of rope around the pole. To serve, a certain player
holds the ball so that its coordinates are (5, 3.6, 4.7) and the coordinates of the end of the rope connected to the pole
are (0, 0, 9.8), where the coordinates are given in feet. Determine the magnitude of the vector representing the
length of the rope.
SOLUTION: Let P represent the point at which the rope and the pole are connected and let B represent the position of the ball as
it is being served. Then
represents the vector from the pole to the ball.
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8-4 Vectors in Three-Dimensional Space
35. TETHERBALL In the game of tetherball, a ball is attached to a 10-foot pole by a length of rope. Two players hit
the ball in opposing directions in attempt to wind the entire length of rope around the pole. To serve, a certain player
holds the ball so that its coordinates are (5, 3.6, 4.7) and the coordinates of the end of the rope connected to the pole
are (0, 0, 9.8), where the coordinates are given in feet. Determine the magnitude of the vector representing the
length of the rope.
SOLUTION: Let P represent the point at which the rope and the pole are connected and let B represent the position of the ball as
it is being served. Then
represents the vector from the pole to the ball.
The magnitude of
is
Thus, the rope is about 8 feet long.
Find each of the following for a =
,b=
, and c =
.
37. 7a – 5b
SOLUTION: 39. 6b + 4c – 4a
SOLUTION: 41. –6a + b + 7c
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8-4 Vectors in Three-Dimensional Space
41. –6a + b + 7c
SOLUTION: Find each of the following for x = –9i + 4j + 3k, y = 6i – 2j − 7k, and z = −2i + 2j + 4k.
43. 3x – 5y + 3z
SOLUTION: Write x, y, and z in component form as
Then rewrite the vector as a linear combination of the standard unit vectors.
45. –8x – 2y + 5z
SOLUTION: Write x, y, and z in component form as
Then rewrite the vector as a linear combination of the standard unit vectors.
47. – x – 4y − z
SOLUTION: Write x, y, and z in component form as
Then rewrite the vector as a linear combination of the standard unit vectors.
49. TRACK AND FIELD Lena throws a javelin due south at a speed of 18 miles per hour and at an angle of 48° relative to the horizontal. If the wind is blowing with a velocity of 12 miles per hour at an angle of S15°E, find a vector that represents the resultant velocity of the javelin. Let i point east, j point north, and k point up.
SOLUTION: Let v be a vector representing the javelin’s velocity and w be a vector representing the wind’s velocity. Since v has
a magnitude of 18 and a direction angle of 48°, find the component form of v using trigonometric ratios as shown.Page 3
Since the direction of the javelin is due south, which is the negative y-axis, the y-component will be negative.
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Then rewrite the vector as a linear combination of the standard unit vectors.
8-4 Vectors in Three-Dimensional Space
49. TRACK AND FIELD Lena throws a javelin due south at a speed of 18 miles per hour and at an angle of 48° relative to the horizontal. If the wind is blowing with a velocity of 12 miles per hour at an angle of S15°E, find a vector that represents the resultant velocity of the javelin. Let i point east, j point north, and k point up.
SOLUTION: Let v be a vector representing the javelin’s velocity and w be a vector representing the wind’s velocity. Since v has
a magnitude of 18 and a direction angle of 48°, find the component form of v using trigonometric ratios as shown.
Since the direction of the javelin is due south, which is the negative y-axis, the y-component will be negative.
Since the wind is blowing with a velocity of 12 miles per hour at an angle of S15°E, find the component form of w
using trigonometric ratios as shown. Since the direction of the wind’s velocity is S15°E, the y-component will be
negative.
The resultant velocity of the javelin is v + w.
If N is the midpoint of
, find P.
51. M (3, 4, 5); N
SOLUTION: Let P = (x2, y 2, z 2). Use the Midpoint Formula and the points M and N to solve for P.
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8-4 Vectors in Three-Dimensional Space
If N is the midpoint of
, find P.
51. M (3, 4, 5); N
SOLUTION: Let P = (x2, y 2, z 2). Use the Midpoint Formula and the points M and N to solve for P.
Solve for x2, y 2, and z 2.
So, P = (4, –2, –1).
53. M (7, 1, 5); N
SOLUTION: Let P = (x2, y 2, z 2). Use the Midpoint Formula and the points M and N to solve for P.
Solve for x2, y 2, and z 2.
So, P = (3, –2, 7).
55. VOLUNTEERING Jody is volunteering to help guide a balloon in a parade. If the balloon is 35 feet high and she is
holding the tether three feet above the ground as shown, how long is the tether to the nearest foot?
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SOLUTION: According to the graph, the coordinates of the balloon are (0, 0, 35) and the coordinates of the end of the tether that
8-4 Vectors in Three-Dimensional Space
So, P = (3, –2, 7).
55. VOLUNTEERING Jody is volunteering to help guide a balloon in a parade. If the balloon is 35 feet high and she is
holding the tether three feet above the ground as shown, how long is the tether to the nearest foot?
SOLUTION: According to the graph, the coordinates of the balloon are (0, 0, 35) and the coordinates of the end of the tether that
Jody is holding are (−4, 10, 3). Use the Distance Formula for points in space to find the length of the tether.
The length of the tether to the nearest foot is 34 feet.
Determine whether the triangle with the given vertices is isosceles or scalene.
57. A(4, 3, 4), B(4, 6, 4), C(4, 3, 6)
SOLUTION: Use the Distance Formula for points in space to find the length of each segment.
The triangle is scalene.
59. A(–2.2, 4.3, 5.6), B(0.7, 9.3, 15.6), C(3.6, 14.3, 5.6)
SOLUTION: Use the Distance Formula for points in space to find the length of each segment.
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8-4 The
Vectors
Three-Dimensional Space
trianglein
is scalene.
59. A(–2.2, 4.3, 5.6), B(0.7, 9.3, 15.6), C(3.6, 14.3, 5.6)
SOLUTION: Use the Distance Formula for points in space to find the length of each segment.
The triangle is isosceles.
61. SPHERES Use the distance formula for two points in space to prove that the standard form of the equation of a 2
2
2
2
sphere with center (h, k, ) and radius r is r = (x − h) + (y − k) + (z − ) .
SOLUTION: Let the center of a sphere with radius r be located at (h, k, ) and the point (x, y, z) be a point on the sphere.
Substitute these points into the distance formula for two points in space. Since the distance between the center of a
sphere and a point on the sphere is equal to the radius, let d = r.
Use the formula r2 = (x − h)2 + (y − k)2 + (z − )2 to write an equation for the sphere with the given
center and radius.
63. center = (6, 0, –1), radius =
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SOLUTION: The formula developed in Exercise 61 for the standard form of the equation of a sphere with a center (h, k, ) and
8-4 Vectors in Three-Dimensional Space
Use the formula r2 = (x − h)2 + (y − k)2 + (z − )2 to write an equation for the sphere with the given
center and radius.
63. center = (6, 0, –1), radius =
SOLUTION: The formula developed in Exercise 61 for the standard form of the equation of a sphere with a center (h, k, ) and
2
2
2
2
radius r is (x − h) + (y − k) + (z − ) = r . Substitute h = 6, k = 0,
= −1, and r =
into the formula.
65. center = (0, 7, –1), radius = 12
SOLUTION: The formula developed in Exercise 61 for the standard form of the equation of a sphere with a center (h, k, ) and
2
2
2
2
radius r is (x − h) + (y − k) + (z − ) = r . Substitute h = 0, k = 7,
= −1, and r = 12 into the formula.
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