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8-5 Dot and Cross Products of Vectors in Space
23. WEIGHTLIFTING A weightlifter doing bicep curls applies 212 newtons of force to lift the dumbbell. The weightlifter’s forearm is 0.356 meters long and she begins the bicep curl with her elbow bent at a 15º angle below
the horizontal, in the direction of the positive x-axis.
a. Find the vector representing the torque about the weightlifter’s elbow in component form.
b. Find the magnitude and direction of the torque.
SOLUTION: a. The component form of the vector representing the directed distance from the axis of rotation to the end of the
weightlifter’s forearm can be found using the triangle shown and trigonometric ratios.
Vector r is therefore
or about . Notice that the z-component is
negative since the weightlifter’s arm is at a 15° angle below the horizontal. The vector representing the force applied
to the weight is 212 newtons up, so
.
Use the cross product of these vectors to find the vector representing the torque about the hinge.
The component form of the torque vector is
.
b. The component form of the torque vector
newton-meters parallel to the negative y-axis.
tells us that the magnitude of the vector is about 72.08 Find the area of the parallelogram with adjacent sides u and v.
25. SOLUTION: First, find
.
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The component form of the torque vector is
.
The component form of the torque vector
tells us that the magnitude of the vector is about 72.08 8-5 b.
Dot
and Cross Products of Vectors in Space
newton-meters parallel to the negative y-axis.
Find the area of the parallelogram with adjacent sides u and v.
25. SOLUTION: First, find
.
Then, find the magnitude of
.
The area of the parallelogram is about 56.7 square units.
27. u = 6i – 2j + 5k , v = 5i – 4j – 8k
SOLUTION: First, write u and v in component form as
Then, find the magnitude of
Next, find
.
.
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8-5 The
Dotarea
and
Cross
Products
of Vectors
Space
of the
parallelogram
is about
56.7 squarein
units.
27. u = 6i – 2j + 5k , v = 5i – 4j – 8k
SOLUTION: First, write u and v in component form as
Then, find the magnitude of
Next, find
.
Next, find
.
.
The area of the parallelogram is about 82.6 square units.
29. u = −3i – 5j + 3k , v = 4i – j + 6k
SOLUTION: First, write u and v in component form as
Then, find the magnitude of
.
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by Cognero
TheManual
area of- Powered
the parallelogram
is about 46.5 square units.
Find the volume of the parallelepiped having t, u, and v as adjacent edges.
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8-5 Dot and Cross Products of Vectors in Space
The area of the parallelogram is about 82.6 square units.
29. u = −3i – 5j + 3k , v = 4i – j + 6k
SOLUTION: First, write u and v in component form as
Then, find the magnitude of
Next, find
.
.
The area of the parallelogram is about 46.5 square units.
Find the volume of the parallelepiped having t, u, and v as adjacent edges.
31. SOLUTION: The volume of the parallelepiped is 206 cubic units.
33. t = −4i + j + 3k , u = 5i + 7j – 6k , v = 3i – 2j – 5k
SOLUTION: Write t, u, and v in component form as
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8-5 The
Dotvolume
and Cross
Productsisof
in Space
of the parallelepiped
206Vectors
cubic units.
33. t = −4i + j + 3k , u = 5i + 7j – 6k , v = 3i – 2j – 5k
SOLUTION: Write t, u, and v in component form as
The volume of the parallelepiped is 102 cubic units.
35. t = 5i – 2j + 6k , u = 3i – 5j + 7k , v = 8i – j + 4k
SOLUTION: Write t, u, and v in component form as
The volume of the parallelepiped is 69 cubic units.
Find a vector that is orthogonal to each vector.
37. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let
and . Find the dot product of a and b.
If a and b are orthogonal, then −x − 2y + 5z = 0.
Substitute values for x and y and solve for z. Let x = 5 and y = 5.
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A vector orthogonal to
Page 5
is .
8-5 Dot and Cross Products of Vectors in Space
The volume of the parallelepiped is 69 cubic units.
Find a vector that is orthogonal to each vector.
37. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let
and . Find the dot product of a and b.
If a and b are orthogonal, then −x − 2y + 5z = 0.
Substitute values for x and y and solve for z. Let x = 5 and y = 5.
A vector orthogonal to
is .
39. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let
and . Find the dot product of a and b.
If a and b are orthogonal, then 7x + 8z = 0.
Substitute a value for x and solve for z. Let x = −8.
A vector orthogonal to
is .
Given v and u · v, find u.
41. v =
, u · v =
SOLUTION: Sample answer: Let
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. Substitute u, v , and
into the equation for a dot product.
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8-5 A
Dot
and Cross Products
of Vectors
in Space
vector orthogonal to
.
is Given v and u · v, find u.
41. v =
, u · v =
SOLUTION: . Substitute u, v , and
Sample answer: Let
into the equation for a dot product.
Substitute a value for x and solve for z. Let x = −1.
Therefore,
.
Determine whether the points are collinear.
43. SOLUTION: Let a = (−1, 7, 7), b = (−3, 9, 11), and c = (−5, 11, 13). Form two vectors,
collinear, then the angle between
and will be 0° or 180°.
Find the component form of each vector.
and . If the three points are
Use the formula for the angle between two vectors.
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.
8-5 Therefore,
Dot and Cross Products
of Vectors in Space
Determine whether the points are collinear.
43. SOLUTION: Let a = (−1, 7, 7), b = (−3, 9, 11), and c = (−5, 11, 13). Form two vectors,
collinear, then the angle between
and will be 0° or 180°.
Find the component form of each vector.
and . If the three points are
Use the formula for the angle between two vectors.
The angle between the vectors is not 180° nor 0°. Therefore, the points are not collinear.
Determine whether each pair of vectors are parallel.
45. SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors.
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8-5 The angle between the vectors is not 180° nor 0°. Therefore, the points are not collinear.
Dot and Cross Products of Vectors in Space
Determine whether each pair of vectors are parallel.
45. SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors.
The angle between the vectors is 0°. Therefore, the vectors are parallel.
47. w =
,
SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors.
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8-5 The angle between the vectors is 0°. Therefore, the vectors are parallel.
Dot and Cross Products of Vectors in Space
47. w =
,
SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors.
The angle between the vectors is 0°. Therefore, the vectors are parallel.
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