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NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 8 5 Glencoe Precalculus

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

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.

b. w = 55 miles per hour at a bearing of S45°E

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. 4.

2.2 cm, 140° 3.5 cm, –12°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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° 2. 2.5 centimeters per hour at a bearing of

from the horizontal 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.

–3.5, 6.1 –1.9, 1.6

16.07 lb; 19.15 lb; 25 lb; 50°

81.93 km/h; 16.8° north of east

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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.

1150 ft at a bearing of 52.1° north of west

44.6 N; 40.1 N 502.49 mph; bearing of 354.3°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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 𝑿𝒀⃑⃑⃑⃑ ⃑ with initial point X(2, –3) and terminal point Y(–4, 2).

Determine the magnitude of 𝑋𝑌⃑⃑⃑⃑ ⃑ using the Distance Formula.

|𝑋𝑌⃑⃑⃑⃑ ⃑| = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)

2

= √(−4 − 2)2 + [2 − (−3)]2

= √(−6)2 + 52

= √61 or about 7.8 units

Represent 𝑋𝑌⃑⃑⃑⃑ ⃑ as an ordered pair.

𝑋𝑌⃑⃑⃑⃑ ⃑ = ⟨𝑥2 – 𝑥1, 𝑦2 – 𝑦1⟩ 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⟩ Substitute.

= ⟨4 + (–1), 2 + 3⟩ or ⟨3, 5⟩ 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 𝑨𝑩⃑⃑⃑⃑⃑⃑ with the given initial and terminal points.

1. A(12, 41), B(52, 33) 2. A(–15, 0), B(7, –19)

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)

⟨40, –8⟩, 8√𝟐𝟔 ⟨22, –19⟩, 13√𝟓

⟨–20, –23⟩ ⟨181, 163⟩

⟨47, 39⟩ ⟨–40, –32⟩

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



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⟩.

u = 1

|𝐯| v Unit vector with the same direction as v

= 1

|⟨−4,−1⟩|⟨–4, –1⟩ Substitute.

= 1

√(−4)2 + (−1)2⟨–4, –1⟩ |⟨𝑎, 𝑏⟩| = √𝑎2 + 𝑏2

= 1

√17⟨–4, –1⟩ Simplify.

= ⟨−4

√17,

−1

√17⟩ or ⟨

−4√17

17,−√17

17⟩ Scalar multiplication

Example 2: Let 𝑴𝑷⃑⃑⃑⃑⃑⃑ ⃑ be the vector with initial point M(2, 2) and terminal point P(5, 4). Write 𝑴𝑷⃑⃑⃑⃑⃑⃑ ⃑ as a linear

combination of the vectors i and j.

First, find the component form of 𝑀𝑃⃑⃑⃑⃑⃑⃑ .

𝑀𝑃⃑⃑⃑⃑⃑⃑ = ⟨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.

𝑀𝑃⃑⃑⃑⃑⃑⃑ = ⟨3, 2⟩ Component form = 3i + 2j ⟨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 𝑀𝑁⃑⃑⃑⃑⃑⃑ ⃑ be the vector with the given initial and terminal points. Write 𝑀𝑁⃑⃑⃑⃑⃑⃑ ⃑ 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 8. ⟨2, 17⟩

⟨𝟒

𝟓,−𝟑

𝟓⟩ ⟨

𝟐√𝟐𝟗

𝟐𝟗,𝟓√𝟐𝟗

𝟐𝟗⟩

–7i – 11j 18i – 2j

⟨–9, –15.6⟩ ⟨–0.44, 5.0⟩

153.4° 83.3°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


8-2 Practice Vectors in the Coordinate Plane

Find the component form and magnitude of 𝑨𝑩⃑⃑⃑⃑⃑⃑ 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⟩

Let 𝑫𝑬⃑⃑⃑⃑⃑⃑ be the vector with the given initial and terminal points. Write 𝑫𝑬⃑⃑⃑⃑⃑⃑ as a linear combination of the

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?

⟨–3, –1⟩; √𝟏𝟎 ⟨1, –3⟩; √𝟏𝟎 ⟨11, 5⟩; √𝟏𝟒𝟔 ⟨6, –3⟩ ⟨–7, 7⟩ ⟨–1, 11⟩ ⟨–21, 28⟩

⟨−√𝟓

𝟓,𝟐√𝟓

𝟓⟩ ⟨−

𝟒√𝟏𝟕

𝟏𝟕, −

√𝟏𝟕

𝟏𝟕⟩

2i – 2j 9i – 5j

–9i – 8j i + 6j

⟨8.9, 8.0⟩ ⟨–5.4, 5.9⟩

295.62 N

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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 = 𝑎1𝑏1 + 𝑎2𝑏2. The vectors a and b are

orthogonal if and only if a ⋅ b = 0. If θ is the angle between nonzero vectors a and b, then cos θ = 𝒂 ⋅ 𝒃

|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⟩

u ⋅ v = 5(–3) + 1(15)

= 0

Since u ⋅ v = 0, u and v are orthogonal.

b. u = ⟨4, 5⟩, v = ⟨8, –6⟩

u ⋅ v = 4(8) + 5(–6)

= 2

Since u ⋅ v ≠ 0, u and v are not orthogonal.

Example 2: Find the angle θ between vectors u and v if u = ⟨5, 1⟩ and v = ⟨–2, 3⟩.

cos θ = 𝒖 ⋅ 𝒗

|𝒖| |𝒗| Angle between two vectors

cos θ = ⟨5,1⟩ ⋅ ⟨−2,3⟩

|⟨5,1⟩| |⟨−2,3⟩| u = ⟨5, 1⟩ and v = ⟨–2, 3⟩

cos θ = −10 + 3

√26 √13 Evaluate.

θ = cos −1 −10 + 3

√26 √13 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⟩ 6. u = 13i – 5j, v = 6i + 2j

0, orthogonal –54, not orthogonal

3√𝟏𝟎 4√𝟏𝟎 179.3° 39.5°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



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:

proj𝐯 u = (𝐮 ⋅ 𝐯

| 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 1 Find the projection of u onto v.

proj𝐯 u = (𝐮 ⋅ 𝐯

| 𝐯|2) v

= ⟨8,6⟩ ⋅ ⟨2,−3⟩

|⟨2,−3⟩|2 ⟨2, –3⟩

= – 2

13 ⟨2, –3⟩ or ⟨−

4

13,

6

13⟩

Step 2 Find u – proj𝐯 u.

= ⟨8, 6⟩ − ⟨−4

13,

6

13⟩

= ⟨108

13,

72

13⟩

Therefore, proj𝐯 u is ⟨−4

13,

6

13⟩ and u = ⟨−

4

13,

6

13⟩ + ⟨

108

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

⟨𝟒𝟎

𝟏𝟕, −

𝟏𝟎

𝟏𝟕⟩; u = ⟨

𝟒𝟎

𝟏𝟕, −

𝟏𝟎

𝟏𝟕⟩ + ⟨

𝟏𝟏

𝟏𝟕,

𝟒𝟒

𝟏𝟕⟩

⟨−𝟑𝟐𝟖

𝟖𝟗, −

𝟐𝟎𝟓

𝟖𝟗⟩; u = ⟨−

𝟑𝟐𝟖

𝟖𝟗, −

𝟐𝟎𝟓

𝟖𝟗⟩ + ⟨−

𝟐𝟗𝟓

𝟖𝟗,

𝟒𝟕𝟐

𝟖𝟗⟩

⟨𝟗

𝟔𝟓, −

𝟕

𝟔𝟓⟩; u = ⟨

𝟗

𝟔𝟓, −

𝟕

𝟔𝟓⟩ + ⟨

𝟓𝟔

𝟔𝟓,

𝟕𝟐

𝟔𝟓⟩

⟨𝟏𝟖𝟎

𝟐𝟗,

𝟏𝟓

𝟐𝟗⟩; u = ⟨

𝟏𝟖𝟎

𝟐𝟗,

𝟏𝟓

𝟐𝟗⟩ + ⟨

𝟐𝟑

𝟐𝟗, −

𝟐𝟕𝟔

𝟐𝟗⟩

⟨−𝟏𝟑𝟐

𝟐𝟎𝟓, −

𝟐𝟖𝟔

𝟐𝟎𝟓⟩; u = ⟨−

𝟏𝟑𝟐

𝟐𝟎𝟓, −

𝟐𝟖𝟔

𝟐𝟎𝟓⟩ + ⟨−

𝟏𝟓𝟎𝟖

𝟐𝟎𝟓,

𝟔𝟗𝟔

𝟐𝟎𝟓⟩

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


8-3 Practice Dot Products and Vector Projections


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.)

0, orthogonal –11, not orthogonal –2, not orthogonal

20.4°

117.9°

109.3°

65.2°

⟨−𝟏𝟐

𝟓,

𝟐𝟒

𝟓⟩; ⟨−

𝟏𝟐

𝟓,

𝟐𝟒

𝟓⟩ + ⟨

𝟑𝟐

𝟓,

𝟏𝟔

𝟓⟩

⟨𝟗𝟗

𝟐, −

𝟑𝟑

𝟐⟩; ⟨

𝟗𝟗

𝟐, −

𝟑𝟑

𝟐⟩ + ⟨

𝟐𝟓

𝟐,

𝟕𝟓

𝟐⟩

⟨−𝟔

𝟐𝟓,

𝟖

𝟐𝟓⟩; ⟨−

𝟔

𝟐𝟓,

𝟖

𝟐𝟓⟩ + ⟨

𝟒𝟒

𝟐𝟓, −

𝟑𝟑

𝟐𝟓⟩

15.8°

300 ft-lb

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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

2,𝑦1 + 𝑦2

2,𝑧1 + 𝑧2

2) = (

10 + 7

2,2 + (−9)

2,−5 + 3

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.

1. (3, 2, 1) 2. (4, −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)

18.06; (𝟓,𝟓

𝟐,𝟓

𝟐)

16.67; (𝟏

𝟐, −𝟕,−

𝟏𝟗

𝟐)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



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⟩ = 𝑣1i + 𝑣2j + 𝑣1k.

Example: Find the component form and magnitude of 𝑨𝑩⃑⃑⃑⃑⃑⃑ with initial point A(–3, 5, 1) and terminal point

B(3, 2, –4). Then find a unit vector in the direction of 𝑨𝑩⃑⃑⃑⃑⃑⃑ .

𝐴𝐵⃑⃑⃑⃑ ⃑ = ⟨𝑥2 – 𝑥1, 𝑦2 – 𝑦1, 𝑧2 – 𝑧1⟩ 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 𝐴𝐵⃑⃑⃑⃑ ⃑ is

|𝐴𝐵⃑⃑ ⃑⃑ ⃑| = √62 + (−3)2 + (−5)2 or √70. 𝐴𝐵⃑⃑⃑⃑ ⃑ = ⟨6, –3, –5⟩

Using this magnitude and component form, find a unit vector u in the direction of 𝐴𝐵⃑⃑⃑⃑ ⃑.

u = 𝐴𝐵⃑⃑ ⃑⃑ ⃑

|𝐴𝐵⃑⃑ ⃑⃑ ⃑| Unit vector in the direction of 𝐴𝐵⃑⃑⃑⃑ ⃑

= ⟨6,−3,−5⟩

√70 or ⟨

3√70

35, −

3√70

70, −

√70

14⟩ 𝐴𝐵⃑⃑⃑⃑ ⃑ = ⟨6, –3, –5⟩ and |𝐴𝐵⃑⃑ ⃑⃑ ⃑| = √70

Exercises

Find the component form and magnitude of 𝑨𝑩⃑⃑⃑⃑⃑⃑ with the given initial and terminal points. Then find a unit vector

in the direction of 𝑨𝑩⃑⃑⃑⃑⃑⃑ .

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 4. –6y + 2z

⟨18, –10, –6⟩, 2√𝟏𝟏𝟓 ⟨9, 8, 17⟩, √𝟒𝟑𝟒;

⟨𝟗√𝟏𝟏𝟓

𝟏𝟏𝟓,√𝟏𝟏𝟓

𝟐𝟑, −

𝟑√𝟏𝟏𝟓

𝟏𝟏𝟓⟩ ⟨

𝟗√𝟒𝟑𝟒

𝟒𝟑𝟒,𝟒√𝟒𝟑𝟒

𝟐𝟏𝟕,𝟏𝟕√𝟒𝟑𝟒

𝟒𝟑𝟒⟩

⟨19, –52, –17⟩ ⟨–10, 54, –34⟩

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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 𝑨𝑩⃑⃑⃑⃑⃑⃑ with the given initial and terminal points.

Then find a unit vector in the direction of 𝑨𝑩⃑⃑⃑⃑⃑⃑ .

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?

⟨–6, 4, 4⟩; 2√𝟏𝟕 ⟨ 3, 1, –9⟩; √𝟗𝟏

⟨−𝟑√𝟏𝟕

𝟏𝟕,𝟐√𝟏𝟕

𝟏𝟕,𝟐√𝟏𝟕

𝟏𝟕⟩ ⟨

𝟑√𝟗𝟏

𝟗𝟏,√𝟗𝟏

𝟗𝟏, −

𝟗√𝟗𝟏

𝟗𝟏⟩

⟨11, –3, –17⟩; √𝟒𝟏𝟗 ⟨1, –11, 17⟩; √𝟒𝟏𝟏

⟨𝟏𝟏√𝟒𝟏𝟗

𝟒𝟏𝟗, −

𝟑√𝟒𝟏𝟗

𝟒𝟏𝟗, −

𝟏𝟕√𝟒𝟏𝟗

𝟒𝟏𝟗⟩ ⟨√

𝟒𝟏𝟏

𝟒𝟏𝟏, −

𝟏𝟏√𝟒𝟏𝟏

𝟒𝟏𝟏,𝟏𝟕√𝟒𝟏𝟏

𝟒𝟏𝟏⟩

√𝟏𝟓𝟖 (−𝟏

𝟐,𝟏𝟏

𝟐, −𝟒) √𝟒𝟒𝟓 (−𝟕, −𝟔,

𝟕

𝟐)

⟨8, –12, 14⟩ ⟨–2, –4, 7⟩

about 143 lb

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


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 = 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3.

The vectors a and b are orthogonal if and only if a ⋅ b = 0.

As with vectors in a plane, if θ is the angle between nonzero vectors a and b, then cos θ = 𝐚 ⋅ 𝐛

|𝐚| |𝐛|

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal.

a. u = ⟨–3, 1, 0⟩, v = ⟨2, 6, 4⟩

u ⋅ v = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3

= −3(2) + 1(6) + 0(4)

= −6 + 6 + 0 or 0

Since u ⋅ v = 0, u and v are orthogonal.

b. u = ⟨3, –2, 1⟩, v = ⟨4, 5, –1⟩

u · v = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3

= 3(4) + (–2)(5) + 1(–1)

= 12 + (–10) –1 or 1

Since u · v ≠ 0, u and v are not orthogonal.

Example 2: Find the angle θ between vectors u and v if u = ⟨4, 8, –3⟩ and v = ⟨9, –3, 0⟩.

cos θ = 𝐮 ⋅ 𝐯

|𝐮| |𝐯| Angle between two vectors

cos θ = ⟨4,8,−3⟩ · ⟨9,−3,0⟩

|⟨4,8,−3⟩| |⟨9,−3,0⟩| u = ⟨4, 8, –3⟩ and v = ⟨9, –3, 0⟩

cos θ = 12

√89 √90 Evaluate the dot product and magnitude.

θ = cos−1 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


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

35; not orthogonal 2, not orthogonal –10, not orthogonal 0, orthogonal

80.0° 41.3°

90.5° 36.8°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



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 = 𝑎1i + 𝑎2j + 𝑎3k and b = 𝑏1i + 𝑏2j + 𝑏3k, the cross product of a and b is the vector

a × b = (𝑎2𝑏3 – 𝑎3𝑏2)i – (𝑎1𝑏3 – 𝑎3𝑏1)j + (𝑎1𝑏2 – 𝑎2𝑏1)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.

u × v = |𝐢 𝐣 𝐤0 4 10 1 3

| u = 0i + 4j + k and v = 0i + j + 3k

= |4 11 3

| i – |0 10 3

|j + |0 40 1

|k Determinant of a 3 × 3 matrix

= (12 – 1)i − (0 – 0)j + (0 − 0)k 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⟩

–14i + 2j – 22k; ⟨–14, 2, –22⟩ ⋅ ⟨2, 3, –1⟩ = –14(2) + 2(3) + (–22)( –1) = 0; ⟨–14, 2, –22⟩ ⋅ ⟨6, –2, –4⟩ = –14(6) + 2(–2) + (–22)( –4) = 0

–8i – 28j + 12k; ⟨–8, –28, 12⟩ ⋅ ⟨5, 2, 8⟩ = –8(5) + (–28)(2) + 12(8) = 0;

⟨–8, –28, 12⟩ ⋅ ⟨–1, 2, 4⟩ = –8(–1) + (–28)(2) + 12(4) = 0

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


8-5 Practice Dot and Cross Products of Vectors in Space


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⟩, 5. u = ⟨3, –2, 1⟩, 6. u = ⟨2, –4, 4⟩, v = ⟨0, 3, –2⟩ v = ⟨–4, –2, 5⟩ 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?

–9; not orthogonal 3; not orthogonal 0; orthogonal about 154.9° about 96.9° about 51.3°

⟨–3, –3, 3⟩; ⟨–3, –3, 3⟩ • ⟨1, 3, 4⟩ ⟨27, 3, 14⟩; ⟨27, 3, 14⟩ • ⟨3, 1, –6⟩ = –3(1) + (–3)(3) + (3)(4) = 0; = (27)(3) + 3(1) + (14)(–6) = 0; ⟨–3, –3, 3⟩ • ⟨–1, 0, –1⟩ ⟨27, 3, 14⟩ • ⟨–2, 4, 3⟩ = (–3)(–1) + (–3)(0) + 3(–1) = 0 = (27)(–2) + (3)(4) + (14)(3) = 0

⟨7, 1, –11⟩; ⟨7, 1, –11⟩ • ⟨3, 1, 2⟩ ⟨1, 4, –7⟩; ⟨1, 4, –7⟩ • ⟨4, –1, 0⟩ = (7)(3) + (1)(1) + (–11)(2) = 0; = (1)(4) + (4)(–1) + (–7)(0) = 0; ⟨7, 1, –11⟩ • ⟨2, –3, 1⟩ = (7)(2) ⟨1, 4, –7⟩ • ⟨5, –3, –1⟩ = (1)(5) + (1)(–3) + (–11)(1) = 0 + (4)(–3) + (–7)(–1) = 0

62.4 units2 74.2 units2

643 units3

8.2 newton meters

8-1 study guide and intervention - mrs. fruge - home · chapter 8 5 glencoe precalculus 8-1 study...

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