NAME _____________________________________________ DATE ____________________________ PERIOD _____________
8-1 Study Guide and Intervention
Introduction to Vectors
Geometric Vectors A vector is a quantity that has both magnitude and direction. The magnitude of a vector is the
length of a directed line segment, and the direction of a vector is the directed angle between the positive x-axis and the
vector. When adding or subtracting vectors, you can use the parallelogram or triangle method to find the resultant.
Example: Use a ruler and a protractor to draw an arrow diagram for each quantity described. Include a scale on
each diagram.
a. v = 60 pounds of force at 125° to the horizontal
b. w = 55 miles per hour at a bearing of S45°E
Using a scale of 1 cm: 20 lb, draw and label a 60 ÷ 20
or 3-centimeter arrow in standard position at a 125°
angle to the x-axis.
Using a scale of 1 cm.: 20 mi/h, draw and label a
55 ÷ 20 or 2.75-centimeter arrow 45° east of south.
Exercises
Use a ruler and a protractor to draw an arrow diagram for each quantity described. Include a scale on each
diagram.
1. r = 30 meters at a bearing of N45°W
2. t = 150 yards at 40° to the horizontal
Find the resultant of each pair of vectors using either the triangle or parallelogram method. State the magnitude of
the resultant in centimeters and its direction relative to the horizontal.
3.
Chapter 8
4.
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8-1 Study Guide and Intervention (continued)
Introduction to Vectors
Vector Applications Vectors can be resolved into horizontal and vertical components.
Example: Suppose Jamal pulls on the ends of a rope tied to a dinghy with a force of 50 Newtons at an angle of 60°
with the horizontal.
a. Draw a diagram that shows the resolution of the force Jamal exerts into its
rectangular components.
Jamal’s pull can be resolved into a horizontal pull x forward and a vertical pull y upward as shown.
b. Find the magnitudes of the horizontal and vertical components of the force.
The horizontal and vertical components of the force form a right triangle. Use the sine or cosine ratios to find the
magnitude of each force.
cos 60° =
|𝒙|
50
Right triangle definitions of cosine and sine
sin 60° =
|𝒚|
50
|𝒙| = 50 cos 60°
Solve for x and y.
|𝒚| = 50 sin 60°
|𝒙| = 25
Use a calculator.
|𝒚| ≈ 43.3
The magnitude of the horizontal component is about 25 Newtons, and the magnitude of the vertical component is
about 43 Newtons.
Exercises
Draw a diagram that shows the resolution of each vector into its rectangular components. Then find the
magnitudes of the vector’s horizontal and vertical components.
1. 7 inches at a bearing of 120°
from the horizontal
2. 2.5 centimeters per hour at a bearing of
N50°W
3. YARDWORK Nadia is pulling a tarp along level ground with a force of 25 pounds directed along the tarp. If the tarp
makes an angle of 50° with the ground, find the horizontal and vertical components of the force. What is the magnitude
and direction of the resultant?
4. TRANSPORTATION A helicopter is moving 15° north of east with a velocity of 52 km/h. If a 30-kilometer per hour
wind is blowing from a bearing of 250°, find the helicopter’s resulting velocity and direction.
Chapter 8
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Glencoe Precalculus
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8-1 Practice
Introduction to Vectors
Use a ruler and a protractor to draw an arrow diagram for each quantity described. Include a scale on each
diagram.
1. r = 60 meters at a bearing of N45°E
2. t = 100 pounds of force at 60° to the
horizontal
3. GROCERY SHOPPING Caroline walks 45° north of west for 1000 feet and then walks 200 feet due north to go
grocery shopping. How far and at what north of west quadrant bearing is Caroline from her apartment?
4. CONSTRUCTION Roland is pulling a crate of construction materials with a force of 60 Newtons at an angle of 42°
with the horizontal.
a. Draw a diagram that shows the resolution of the force Roland exerts into its rectangular components.
b. Find the magnitudes of the horizontal and vertical components of the force.
5. AVIATION An airplane is flying with an airspeed of 500 miles per hour on a heading due north. If a 50-mile per hour
wind is blowing at a bearing of 270°, determine the velocity and direction of the plane relative to the ground.
Chapter 8
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8-2 Study Guide and Intervention
Vectors in the Coordinate Plane
Vectors in the Coordinate Plane The magnitude of a vector in the coordinate plane is found using the Distance
Formula.
Example 1: Find the magnitude of ⃑⃑⃑⃑⃑
𝑿𝒀 with initial point X(2, –3) and terminal point Y(–4, 2).
⃑⃑⃑⃑⃑ using the Distance Formula.
Determine the magnitude of 𝑋𝑌
⃑⃑⃑⃑⃑ | = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
|𝑋𝑌
= √(−4 − 2)2 + [2 − (−3)]2
= √(−6)2 + 52
= √61 or about 7.8 units
⃑⃑⃑⃑⃑ as an ordered pair.
Represent 𝑋𝑌
⃑⃑⃑⃑⃑ = ⟨𝑥2 – 𝑥1 , 𝑦2 – 𝑦1⟩
𝑋𝑌
Component form
= ⟨–4 – 2, 2 – (–3)⟩
(𝑥1 , 𝑦1 ) = (2, –3) and (𝑥2 , 𝑦2 ) = (–4, 2)
= ⟨–6, 5⟩
Subtract.
Example 2: Find each of the following for s = ⟨4, 2⟩ and t = ⟨–1, 3⟩.
a. s + t
s + t = ⟨4, 2⟩ + ⟨–1, 3⟩
= ⟨4 + (–1), 2 + 3⟩ or ⟨3, 5⟩
Substitute.
Vector addition
b. 3s + t
3s + t = 3⟨4, 2⟩ + ⟨–1, 3⟩
Substitute.
= ⟨12, 6⟩ + ⟨–1, 3⟩
Scalar multiplication
= ⟨11, 9⟩
Vector addition
Exercises
Find the component form and magnitude of the vector ⃑⃑⃑⃑⃑⃑
𝑨𝑩 with the given initial and terminal points.
2. A(–15, 0), B(7, –19)
1. A(12, 41), B(52, 33)
Find each of the following for f = ⟨4, –2⟩, g = ⟨24, 21⟩, and h = ⟨–1, –3⟩.
3. f – g
4. 8g – 2f + 3h
5. 2g + h
6. f – 2(g + 2h)
Chapter 8
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8-2 Study Guide and Intervention (continued)
Vectors in the Coordinate Plane
Unit Vectors A vector that has a magnitude of 1 unit is called a unit vector. A unit vector in the direction of the positive
x-axis is denoted as i = ⟨1, 0⟩, and a unit vector in the direction of the positive y-axis is denoted as j = ⟨0, 1⟩. Vectors can
be written as linear combinations of unit vectors by first writing the vector as an ordered pair and then writing it as a sum
of the vectors i and j.
Example 1: Find a unit vector u with the same direction as v = ⟨–4, –1⟩.
1
u = |𝐯| v
Unit vector with the same direction as v
1
= |⟨−4,−1⟩|⟨–4, –1⟩
=
=
1
√(−4)2 + (−1)2
1
Substitute.
⟨–4, –1⟩
|⟨𝑎, 𝑏⟩| = √𝑎2 + 𝑏2
⟨–4, –1⟩
Simplify.
√17
=⟨
−4
,
−1
⟩ or ⟨
√17 √17
−4√17 −√17
17
,
17
⟩
Scalar multiplication
⃑⃑⃑⃑⃑⃑⃑ be the vector with initial point M(2, 2) and terminal point P(5, 4). Write 𝑴𝑷
⃑⃑⃑⃑⃑⃑⃑ as a linear
Example 2: Let 𝑴𝑷
combination of the vectors i and j.
⃑⃑⃑⃑⃑⃑ .
First, find the component form of 𝑀𝑃
⃑⃑⃑⃑⃑⃑
𝑀𝑃 = ⟨𝒙𝟐 – 𝒙𝟏, 𝒚𝟐 – 𝒚𝟏⟩
Component form
= ⟨5 – 2, 4 – 2⟩ or ⟨3, 2⟩
(𝑥1 , 𝑦1 ) = (2, 2) and (𝑥2 , 𝑦2 ) = (5, 4)
Then rewrite the vector as a linear combination of the standard unit vectors.
⃑⃑⃑⃑⃑⃑ = ⟨3, 2⟩
𝑀𝑃
= 3i + 2j
Component form
⟨a, b⟩ = ai + bj
Exercises
Find a unit vector u with the same direction as the given vector.
1. p = ⟨4, –3⟩
2. w = ⟨10, 25⟩
Let ⃑⃑⃑⃑⃑⃑⃑
𝑀𝑁 be the vector with the given initial and terminal points. Write ⃑⃑⃑⃑⃑⃑⃑
𝑀𝑁 as a linear combination of the
vectors i and j.
3. M(2, 8), N(–5, –3)
4. M(0, 6), N(18, 4)
Find the component form of v with the given magnitude and direction angle.
5. |v| = 18, θ = 240°
6. |v| = 5, θ = 95°
Find the direction angle of each vector to the nearest tenth.
7. –4i + 2j
Chapter 8
8. ⟨2, 17⟩
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8-2 Practice
Vectors in the Coordinate Plane
Find the component form and magnitude of ⃑⃑⃑⃑⃑⃑
𝑨𝑩 with the given initial and terminal points.
1. A(2, 4), B(–1, 3)
2. A(4, –2), B(5, –5)
3. A(–3, –6), B(8, –1)
Find each of the following for v = ⟨2, –1⟩ and w = ⟨–3, 5⟩ .
4. 3v
5. w – 2v
6. 4v + 3w
7. 5w – 3v
Find a unit vector u with the same direction as v.
8. v = ⟨–3, 6⟩
9. v = ⟨–8, –2⟩
⃑⃑⃑⃑⃑⃑ be the vector with the given initial and terminal points. Write 𝑫𝑬
⃑⃑⃑⃑⃑⃑ as a linear combination of the
Let 𝑫𝑬
vectors i and j.
10. D(4, –5), E(6, –7)
11. D(–4, 3), E(5, –2)
12. D(4, 6), E(–5, –2)
13. D(2, 1), E(3, 7)
Find the component form of v with the given magnitude and direction angle.
14. |v| = 12, θ = 42°
15. |v| = 8, θ = 132°
16. GARDENING Anne and Henry are lifting a stone statue and moving it to a new location in their garden. Anne is
pushing the statue with a force of 120 newtons at a 60° angle with the horizontal while Henry is pulling the statue with
a force of 180 newtons at a 40° angle with the horizontal. What is the magnitude of the combined force they exert on
the statue?
Chapter 8
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8-3 Study Guide and Intervention
Dot Products and Vector Projections
Dot Product The dot product of a = ⟨𝑎1 , 𝑎2 ⟩ and b = ⟨𝑏1, 𝑏2⟩ is defined as a ⋅ b = 𝑎1 𝑏1 + 𝑎2 𝑏2. The vectors a and b are
𝒂⋅𝒃
orthogonal if and only if a ⋅ b = 0. If θ is the angle between nonzero vectors a and b, then cos θ = |𝒂| |𝒃|.
Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal.
a. u = ⟨5, 1⟩, v = ⟨–3, 15⟩
b. u = ⟨4, 5⟩, v = ⟨8, –6⟩
u ⋅ v = 5(–3) + 1(15)
u ⋅ v = 4(8) + 5(–6)
=0
=2
Since u ⋅ v ≠ 0, u and v are not orthogonal.
Since u ⋅ v = 0, u and v are orthogonal.
Example 2: Find the angle θ between vectors u and v if u = ⟨5, 1⟩ and v = ⟨–2, 3⟩.
𝒖⋅𝒗
cos θ = |𝒖| |𝒗|
Angle between two vectors
⟨5,1⟩ ⋅ ⟨−2,3⟩
cos θ = |⟨5,1⟩| |⟨−2,3⟩|
cos θ =
u = ⟨5, 1⟩ and v = ⟨–2, 3⟩
−10 + 3
√26 √13
−1 −10 + 3
θ = cos
√26 √13
Evaluate.
or about 112°
Simplify and solve for θ.
The measure of the angle between u and v is about 112°.
Exercises
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = ⟨2, 4⟩, v = ⟨–12, 6⟩
2. u = –8i + 5j, v = 3i –6j
Use the dot product to find the magnitude of the given vector.
3. a = ⟨9, 3⟩
4. c = ⟨–12, 4⟩
Find the angle θ between u and v to the nearest tenth of a degree.
5. u = ⟨–3, –5⟩, v = ⟨7, 12⟩
Chapter 8
6. u = 13i – 5j, v = 6i + 2j
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8-3 Study Guide and Intervention (continued)
Dot Products and Vector Projections
Vector Projection A vector projection is the decomposition of a vector u into two perpendicular parts, 𝐰𝟏 and 𝐰𝟐 ,
in which one of the parts is parallel to another vector v. When you find the projection of u onto v, you are finding a
component of u that is parallel to v. To find the projection of u onto v, use the formula:
𝐮⋅𝐯
proj𝐯 u = (| 𝐯|2 ) v.
Example: Find the projection of u = ⟨8, 6⟩ onto v = ⟨2, –3⟩ . Then write u as the sum of two orthogonal vectors,
one of which is the projection of u onto v.
Step 2 Find u – proj𝐯 u.
Step 1 Find the projection of u onto v.
proj𝐯 u = (
=
𝐮⋅𝐯
| 𝐯|2
⟨8,6⟩ ⋅ ⟨2,−3⟩
=–
|⟨2,−3⟩|2
2
13
4
=⟨
⟨2, –3⟩
⟨2, –3⟩ or ⟨−
4
6
= ⟨8, 6⟩ − ⟨− 13 , 13⟩
)v
6
4
,
6
13 13
108
13
,
72
13
⟩
⟩
4
6
108
Therefore, proj𝐯 u is ⟨− 13 , 13⟩ and u = ⟨− 13 , 13⟩ + ⟨ 13 ,
72
13
⟩.
Exercises
Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is the
projection of u onto v.
1. u = ⟨3, 2⟩ , v = ⟨–4, 1⟩
2. u = ⟨−7, 3⟩ , v = ⟨8, 5⟩
3. u = ⟨1, 1⟩ , v = ⟨9, –7⟩
4. u = 7i – 9j, v = 12i + j
5. u = −8i + 2j, v = 6i + 13j
Chapter 8
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8-3 Practice
Dot Products and Vector Projections
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = ⟨3, 6⟩ , v = ⟨–4, 2⟩
2. u = –i + 4j, v = 3i – 2j
3. u = ⟨2, 0⟩ , v = ⟨–1, –1⟩
Find the angle θ between u and v to the nearest tenth of a degree.
4. u = ⟨–1, 9⟩, v = ⟨3, 12⟩
5. u = ⟨–6, –2⟩, v = ⟨2, 12⟩
6. u = 27i + 14j, v = i – 7j
7. u = 5i – 4j, v = 2i + j
Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is the
projection of u onto v.
8. u = ⟨4, 8⟩ , v = ⟨–1, 2⟩
9. u = ⟨62, 21⟩ , v = ⟨–12, 4⟩
10. u = ⟨–2, –1⟩ , v = ⟨–3, 4⟩
11. TRANSPORTATION Train A and Train B depart from the same station. The path that train A takes can be
represented by ⟨33, 12⟩. If the path that train B takes can be represented by ⟨55, 4⟩, find the angle between the
pair of vectors.
12. PHYSICS Janna is using a force of 100 pounds to push a cart up a ramp. The ramp is 6 feet long and is at a 30° angle
with the horizontal. How much work is Janna doing in the vertical direction? (Hint: Use the sine ratio and the formula
W = F ⋅ d.)
Chapter 8
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8-4 Study Guide and Intervention
Vectors in Three-Dimensional Space
Coordinates in Three Dimensions Ordered triples, like ordered pairs, can be used to represent vectors.
Operations on vectors represented by ordered triples are similar to those on vectors represented by ordered pairs.
Example: HIKING The location of two hikers are represented by the coordinates (10, 2, –5) and (7, –9, 3),
where the coordinates are given in kilometers.
a. How far apart are the hikers?
Use the Distance Formula for points in space.
AB = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2
Distance Formula
= √(7 − 10)2 + ((−9) − 2)2 + (3 − (−5))2
(𝑥1 , 𝑦1 , 𝑧1 ) = (10, 2, –5) and
≈ 13.93
(𝑥2 , 𝑦2 , 𝑧2 ) = (7, –9, 3)
The hikers are about 14 kilometers apart.
b. The hikers decided to meet at the midpoint between their paths. What are the coordinates of the midpoint?
Use the Midpoint Formula for points in space.
(
𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑧1 + 𝑧2
2
,
2
,
2
)=(
10 + 7 2 + (−9) −5 + 3
2
,
2
,
2
)
(𝑥1 , 𝑦1 , 𝑧1 ) = (10, 2, –5) and
≈ (8.5, –3.5, –1)
(𝑥2 , 𝑦2 , 𝑧2 ) = (7, –9, 3)
The midpoint is at the coordinates (8.5, –3.5, –1).
Exercises
Plot each point in a three-dimensional coordinate system.
2. (4, −2, –1)
1. (3, 2, 1)
Find the length and midpoint of the segment with the given endpoints.
3. (8, –3, 9), (2, 8, –4)
4. (–6, –12, –8), (7, –2, –11)
Chapter 8
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8-4 Study Guide and Intervention (continued)
Vectors in Three-Dimensional Space
Vectors in Space Operations on vectors represented by ordered triples are similar to those on vectors represented by
ordered pairs. Three-dimensional vectors can be added, subtracted, and multiplied by a scalar in the same ways. In space,
a vector v in standard position with a terminal point located at (𝑣1 , 𝑣2 , 𝑣3 ) is denoted by ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩. Thus, the zero
vector is 0 = ⟨0, 0, 0⟩ and the standard unit vectors are i = ⟨1, 0, 0⟩, j = ⟨0, 1, 0⟩, and k = ⟨0, 0, 1⟩. The component form
of v can be expressed as a linear combination of these unit vectors, ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩ = 𝑣1 i + 𝑣2 j + 𝑣1 k.
Example: Find the component form and magnitude of ⃑⃑⃑⃑⃑⃑
𝑨𝑩 with initial point A(–3, 5, 1) and terminal point
B(3, 2, –4). Then find a unit vector in the direction of ⃑⃑⃑⃑⃑⃑
𝑨𝑩.
⃑⃑⃑⃑⃑ = ⟨𝑥2 – 𝑥1 , 𝑦2 – 𝑦1, 𝑧2 – 𝑧1 ⟩
𝐴𝐵
Component form of vector
= ⟨3 – (–3), 2 – 5, –4 – 1⟩ or ⟨6, –3, –5⟩
(𝑥1 , 𝑦1 , 𝑧1 ) = (–3, 5, 1) and (𝑥2 , 𝑦2 , 𝑧2 ) = (3, 2, –4)
Using the component form, the magnitude of ⃑⃑⃑⃑⃑
𝐴𝐵 is
⃑⃑⃑⃑⃑ | = √62 + (−3)2 + (−5)2 or √70.
|𝐴𝐵
⃑⃑⃑⃑⃑ = ⟨6, –3, –5⟩
𝐴𝐵
Using this magnitude and component form, find a unit vector u in the direction of ⃑⃑⃑⃑⃑
𝐴𝐵.
⃑⃑⃑⃑⃑
𝐴𝐵
⃑⃑⃑⃑⃑
Unit vector in the direction of 𝐴𝐵
u = |𝐴𝐵
⃑⃑⃑⃑⃑ |
=
⟨6,−3,−5⟩
√70
or ⟨
3√70
35
,−
3√70
70
,−
√70
⟩
14
⃑⃑⃑⃑⃑
⃑⃑⃑⃑⃑ | = √70
𝐴𝐵 = ⟨6, –3, –5⟩ and |𝐴𝐵
Exercises
Find the component form and magnitude of ⃑⃑⃑⃑⃑⃑
𝑨𝑩 with the given initial and terminal points. Then find a unit vector
in the direction of ⃑⃑⃑⃑⃑⃑
𝑨𝑩.
1. A(–10, 3, 9), B(8, –7, 3)
2. A(–1, –4, –7), B(8, 4, 10)
Find each of the following for x = 3i + 2j – 5k, y = i – 5j + 7k, and z = –2i + 12j + 4k.
3. 3x + 2y – 4z
Chapter 8
4. –6y + 2z
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8-4 Practice
Vectors in Three-Dimensional Space
Plot each point in a three-dimensional coordinate system.
1. (–3, 4, –1)
2. (2, 0, –5)
Locate and graph each vector in space.
3. ⟨4, 7, 6⟩
4. ⟨4, –2, 6⟩
Find the component form and magnitude of ⃑⃑⃑⃑⃑⃑
𝑨𝑩 with the given initial and terminal points.
Then find a unit vector in the direction of ⃑⃑⃑⃑⃑⃑
𝑨𝑩 .
5. A(2, 1, 3), B(–4, 5, 7)
6. A(4, 0, 6), B(7, 1, –3)
7. A(–4, 5, 8), B(7, 2, –9)
8. A(6, 8, –5), B(7, –3, 12)
Find the length and midpoint of the segment with the given endpoints.
9. (3, 4, –9), (–4, 7, 1)
10. (–17, –3, 2), (3, –9, 5)
Find each of the following for v = ⟨2, –4, 5⟩ and w = ⟨6, –8, 9⟩.
11. v + w
12. 5v – 2w
13. PHYSICS Suppose that the force acting on an object can be expressed by the vector ⟨85, 35, 110⟩, where each
measure in the ordered triple represents the force in pounds. What is the magnitude of this force?
Chapter 8
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8-5 Study Guide and Intervention
Dot and Cross Products of Vectors in Space
Dot Products in Space The dot product of two vectors in space is an extension of the dot product
of two vectors in a plane. Similarly, the dot product of two vectors is a scalar. The dot product of
a = ⟨𝑎1 , 𝑎2 , 𝑎3 ⟩ and b = ⟨𝑏1, 𝑏2, 𝑏3⟩ is defined as a ∙ b = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3.
The vectors a and b are orthogonal if and only if a ⋅ b = 0.
𝐚⋅𝐛
As with vectors in a plane, if θ is the angle between nonzero vectors a and b, then cos θ = |𝐚| |𝐛|
Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal.
b. u = ⟨3, –2, 1⟩, v = ⟨4, 5, –1⟩
a. u = ⟨–3, 1, 0⟩, v = ⟨2, 6, 4⟩
u ⋅ v = 𝑢1 𝑣1 + 𝑢2 𝑣2 + 𝑢3 𝑣3
u · v = 𝑢1 𝑣1 + 𝑢2 𝑣2 + 𝑢3 𝑣3
= −3(2) + 1(6) + 0(4)
= 3(4) + (–2)(5) + 1(–1)
= −6 + 6 + 0 or 0
= 12 + (–10) –1 or 1
Since u · v ≠ 0, u and v are not orthogonal.
Since u ⋅ v = 0, u and v are orthogonal.
Example 2: Find the angle θ between vectors u and v if u = ⟨4, 8, –3⟩ and v = ⟨9, –3, 0⟩.
𝐮⋅𝐯
cos θ = |𝐮| |𝐯|
Angle between two vectors
⟨4,8,−3⟩ · ⟨9,−3,0⟩
4,8,−3⟩| |⟨9,−3,0⟩|
cos θ = |⟨
cos θ =
12
Evaluate the dot product and magnitude.
√89 √90
θ = cos −1
u = ⟨4, 8, –3⟩ and v = ⟨9, –3, 0⟩
12
89.5
or about 82.3°
Simplify and solve for θ.
The measure of the angle between u and v is about 82.3°.
Exercises
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = ⟨3, –2, 9⟩ , v = ⟨1, 2, 4⟩
2. u = ⟨–2, –4, –6⟩ , v = ⟨–3, 7, –4⟩
3. u = ⟨4, –3, 8⟩ , v = ⟨2, –2, –3⟩
4. u = 3i + 6j – 3k, v = –5i – 2j – 9k
Find the angle θ between vectors u and v to the nearest tenth of a degree.
5. u = ⟨5, –22, 9⟩ , v = ⟨14, 2, 4⟩
6. u = ⟨4, –5, 7⟩ , v = ⟨11, –8, 2⟩
7. u = –4i + 5j – 3k, v = –8i – 12j – 9k
8. u = i + 2j – k, v = –i + 4j – 3k
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8-5 Study Guide and Intervention (continued)
Dot and Cross Products of Vectors in Space
Cross Products Unlike the dot product, the cross product of two vectors is a vector. This vector does not lie in the
plane of the given vectors but is perpendicular to the plane containing the two vectors.
Cross Product of Vectors in Space
If a = 𝑎1 i + 𝑎2 j + 𝑎3 k and b = 𝑏1i + 𝑏2j + 𝑏3k, the cross product of a and b is the vector
a × b = (𝑎2 𝑏3 – 𝑎3 𝑏2)i – (𝑎1 𝑏3 – 𝑎3 𝑏1)j + (𝑎1 𝑏2 – 𝑎2 𝑏1)k.
If two vectors have the same initial point and form the sides of a parallelogram, the magnitude of the cross product
will give you the area of the parallelogram. If three vectors have the same initial point and form adjacent edges of a
parallelepiped, then the absolute value of the triple scalar product gives the volume. To find the triple scalar product,
use the same matrix set up that is used for cross products, but i, j, and k are replaced by the third vector.
Example : Find the cross product of u = ⟨0, 4, 1⟩ and v = ⟨0, 1, 3⟩. Then show that u × v is orthogonal to
both u and v.
𝐢 𝐣 𝐤
u × v = |0 4 1|
0 1 3
4 1
0 1
0 4
|i–|
|j + |
|k
=|
1 3
0 3
0 1
= (12 – 1)i − (0 – 0)j + (0 − 0)k
u = 0i + 4j + k and v = 0i + j + 3k
Determinant of a 3 × 3 matrix
Determinants of 2 × 2 matrices
= 11i – 0j + 0k
Simplify.
= 11i or ⟨11, 0, 0⟩
Component form
To show that u × v is orthogonal to both u and v, find the dot product of u × v with u and u × v with v.
(u × v) ⋅ u
(u × v) ⋅ v
= ⟨11, 0, 0⟩ ⋅ ⟨0, 4, 1⟩
= ⟨11, 0, 0⟩ ⋅ ⟨0, 1, 3⟩
= 11(0) + 0(4) + 0(1)
= 11(0) + 0(1) + 0(3)
=0+0+0
=0+0+0
=0✓
=0✓
Because both dot products are zero, the vectors are orthogonal.
Exercises
Find the cross product of u and v. Then show that u × v is orthogonal to both u and v.
1. u = ⟨2, 3, –1⟩, v = ⟨6, –2, –4⟩
2. u = ⟨5, 2, 8⟩, v = ⟨–1, 2, 4⟩
Chapter 8
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
8-5 Practice
Dot and Cross Products of Vectors in Space
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. ⟨–2, 0, 1⟩ ⋅ ⟨3, 2, –3⟩
2. ⟨–4, –1, 1⟩ ⋅ ⟨1, –3, 4⟩
3. ⟨0, 0, 1⟩ ⋅ ⟨1, –2, 0⟩
Find the angle θ between vectors u and v to the nearest tenth of a degree.
4. u = ⟨1, –2, 1⟩,
v = ⟨0, 3, –2⟩
5. u = ⟨3, –2, 1⟩,
v = ⟨–4, –2, 5⟩
6. u = ⟨2, –4, 4⟩,
v = ⟨–2, –1, 6⟩
Find the cross product of u and v. Then show that u × v is orthogonal to both u and v.
7. ⟨1, 3, 4⟩ × ⟨–1, 0, –1⟩
8. ⟨3, 1, –6⟩ × ⟨–2, 4, 3⟩
9. ⟨3, 1, 2⟩ × ⟨2, –3, 1⟩
10. ⟨4, –1, 0⟩ × ⟨5, –3, –1⟩
Find the area of the parallelogram with adjacent sides u and v.
11. u = ⟨9, 4, 2⟩ , v = ⟨6, –4, 2⟩
12. u = ⟨2, 0, –8⟩ , v = ⟨–3, –8, –5⟩
13. Find the volume of the parallelepiped with adjacent edges represented by the vectors ⟨3, –2, 9⟩, ⟨6, –2, –7⟩,
and ⟨–8, –5, –2⟩ .
14. TOOLS A mechanic applies a force of 35 newtons straight down to a ratchet that is 0.25 meter long.
What is the magnitude of the torque when the handle makes a 20° angle above the horizontal?
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