1. Five-Minute Check (over Lesson 8-2) Then/Now New Vocabulary
Key Concept: Dot Product of Vectors in a Plane
Key Concept: Orthogonal Vectors
Example 1: Find the Dot Product to Determine Orthogonal Vectors
Key Concept and Proof: Properties of the Dot Product
Example 2: Use the Dot Product to Find Magnitude
Key Concept and Proof: Angle Between Two Vectors
Example 3: Find the Angle Between Two Vectors
Key Concept and Proof: Projection of u onto v
Example 4: Find the Projection of u onto v
Example 5: Projection with Direction Opposite v
Example 6: Real-World Example: Use a Vector Projection to Find a Force
Example 7: Real-World Example: Calculate Work
Lesson Menu

2. Find the component form and magnitude of with initial point A(−3, 7) and terminal point B(6, 2).
5–Minute Check 1

3. Find the component form and magnitude of with initial point A(2, 5) and terminal point B(8, –3).
5–Minute Check 2

4. Find 2f − 3g + 4h if f = −7, −2, g = 3, 1, and h = 9, −1.
B. –11, 13
C. 41, –11
D. 41, 0
5–Minute Check 3

5. Find a unit vector with the same direction as v = −1, 5.B. C.
5–Minute Check 4

6. Which of the following represents the direction angle of the vector 2i − 8j?
B °
C °
D °
5–Minute Check 5

7. Find the projection of one vector onto another.
You found the magnitudes of and operated with algebraic vectors. (Lesson 8-2)
Find the dot product of two vectors and use the dot product to find the angle between them.
Find the projection of one vector onto another.
Then/Now

8. dot product
orthogonal
vector projection
work
Vocabulary

9. Key Concept 1

10. Key Concept 1

11. Since u ● v ≠ 0, u and v are not orthogonal, as illustrated.
Find the Dot Product to Determine Orthogonal Vectors
A. Find the dot product of u and v if u = –3, 4 and v = 3, 6. Then determine if u and v are orthogonal.
u ● v = –3(3) + 4(6)
= 15
Since u ● v ≠ 0, u and v are not orthogonal, as illustrated.
Answer: 15; not orthogonal
Example 1

12. Since u ● v = 0, u and v are orthogonal, as illustrated.
Find the Dot Product to Determine Orthogonal Vectors
B. Find the dot product of u and v if u = 2, 7 and v = –14, 4 . Then determine if u and v are orthogonal.
u ● v = 2(–14) + 7(4)
= 0
Since u ● v = 0, u and v are orthogonal, as illustrated.
Answer: 0; orthogonal
Example 1

13. Find the dot product of u = 4, –1 and v = –3, –5
Find the dot product of u = 4, –1 and v = –3, –5. Then determine if u and v are orthogonal.
A. –7; yes
B. –7; no
C. 17; yes
D. 17; no
Example 1

14. Key Concept 2

15. Use the dot product to find the magnitude of a = –6, 5.
Use the Dot Product to Find Magnitude
Use the dot product to find the magnitude of a = –6, 5.
Since |a|2 = a ● a, then |a| =
a = –6, 5
Simplify.
The magnitude of a is or about 7.81.
Answer: or about 7.81
Example 2

16. Use the dot product to find the magnitude of v = –4, –1.
B. 17 C.
D. 15
Example 2

17. Key Concept 4

18. Angle between two vectors
Find the Angle Between Two Vectors
A. Find the angle θ between u = –3, –5 and v = 2, –3 to the nearest tenth of a degree.
Angle between two vectors
u = –3, –5 and v = 2, –3
Evaluate.
Simplify.
Example 3

19. The measure of the angle between u and v is about 64.7°.
Find the Angle Between Two Vectors
Solve for θ.
The measure of the angle between u and v is about 64.7°.
Answer: 64.7°
Example 3

20. Angle between two vectors
Find the Angle Between Two Vectors
B. Find the angle θ between u = 1, –4 and v = 2, 6 to the nearest tenth of a degree.
Angle between two vectors
u = 1, –4 and v = 2, 6
Evaluate.
Simplify.
Example 3

21. The measure of the angle between u and v is about 147.5°.
Find the Angle Between Two Vectors
Solve for .
The measure of the angle between u and v is about 147.5°.
Answer: °
Example 3

22. Find the angle between vectors v = –3, –1 and w = –5, 2 to the nearest tenth of a degree.
Example 3

23. Key Concept 3

24. Step 1 Find the projection of u onto v.
Find the projection of u = –1, 5 onto v = 4, 6. 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.
Projection of u onto v
u = –1, 5 onto v = 4, 6
Evaluate.
Example 4

25. Scalar multiplication
Find the Projection of u onto v.
Scalar multiplication
Step 2 Find w2.
Since u = w1 + w2, w2 = u – w1.
w2 = u – w1
= u – projvu
= –1, 5 – 2, 3
= –3, 2
Therefore, projvu is w1 = 2, 3 as shown on the next slide, and u = 2, 3 + –3, 2 .
Example 4

26. Answer: projvu = 2, 3; u = 2, 3 + –3, 2
Find the Projection of u onto v.
Answer: projvu = 2, 3; u = 2, 3 + –3, 2
Example 4

27. Find the projection of u = 4, –3 onto v = 1, 1
Find the projection of u = 4, –3 onto v = 1, 1. Then write u as a sum of two orthogonal vectors, one of which is the projection of u onto v.
A. projvu =
B. projvu =
C. projvu =
D. projvu =
Example 4

28. Projection with Direction Opposite v
Find the projection of u = 4, 2 onto v = –3, 5. Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v.
Notice that the angle between u and v is obtuse, so the projection of u onto v lies on the vector opposite v or –v, as shown above.
Example 5

29. Step 1 Find the projection of u onto v.
Projection with Direction Opposite v
Step 1 Find the projection of u onto v.
Projection of u onto v
u = 4, 2 onto v = –3, 5
Evaluate.
Scalar multiplication
Example 5

30. Step 2 Find w2. Since u = w 1 + w2, w2 = u – w 1. w2 = u – w 1
Projection with Direction Opposite v
Step 2 Find w2.
Since u = w 1 + w2, w2 = u – w 1.
w2 = u – w 1
= u – projvu
Example 5

31. Therefore, projvu is w1 as shown, and u .
Projection with Direction Opposite v
Therefore, projvu is w as shown, and u
Answer:
Example 5

32. Find the projection of u = 3, –2 onto v = 4, 3
Find the projection of u = 3, –2 onto v = 4, 3. Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v.
A. projvu =
B. projvu =
C. projvu =
D. projvu =
Example 5

33. Use a Vector Projection to Find a Force
BOULDERS A 10,000-pound boulder sits on a mountain at an incline of 60°. Ignoring the force of friction, what force is required to keep the boulder from rolling down the mountain?
The weight of the boulder is the force exerted due to gravity, F = 0, –10,000. To find the force –w1 required to keep the boulder from rolling down the mountain, project F onto a unit vector v in the direction of the side of the mountain.
Example 6

34. Step 1 Find a unit vector v in the direction of the mountain.
Use a Vector Projection to Find a Force
Step 1 Find a unit vector v in the direction of the mountain.
v = |v| (cos θ), |v| (sin θ) Component form of v in terms of |v| and θ
Example 6

35. = 1(cos 60°), 1(sin 60°) or |v| = 1 and θ = 60°
Use a Vector Projection to Find a Force
= 1(cos 60°), 1(sin 60°) or |v| = 1 and
θ = 60°
Step 2 Find w1, the projection of F onto unit vector v, projvF.
Projection of F onto v
Since v is a unit vector, |v| = 1. Simplify.
Example 6

36. F = 0, –10,000 and v = Find the dot product.
Use a Vector Projection to Find a Force
F = 0, –10,000 and v =
Find the dot product.
The force required is –w1 = –(8660.3v) or v. Since v is a unit vector, this means that this force has a magnitude of about pounds and is in the direction of the side of the mountain.
Answer: about lb
Example 6

37. TRUCKS A 5000-pound truck sits on a hill inclined at a 15° angle
TRUCKS A 5000-pound truck sits on a hill inclined at a 15° angle. Ignoring the force of friction, what force is required to keep the truck from rolling down the hill?
A lb
B lb
C lb
D lb
Example 6

38. Calculate Work
MOWING A person pushes a reel mower with a constant force of 40 newtons at a constant angle of 45°. Find the work done in joules moving the mower 12 meters.
Example 7

39. Method 1 Use the projection formula for work.
Calculate Work
Method 1 Use the projection formula for work.
The magnitude of the projection of F onto is |F| cos θ = 40 cos 45°. The magnitude of , the directed distance, is 12.
W = Projection formula for work
= (40 cos 45°)(12) Substitution or about 339.4
Example 7

40. Method 2 Use the dot product formula for work.
Calculate Work
Method 2 Use the dot product formula for work.
The component form of the force vector F in terms of magnitude and direction angle given is 40 cos (–45°), 40 sin (–45°). The component form of the directed distance the mower is moved is 12, 0.
W = F ● Dot product formula for work
= 40 cos (–45°), 40 sin (–45°) ● 12, 0 Substitution
= [40 cos (–45°)](12) or about Dot product
Therefore, the person does about joules of work pushing the mower.
Answer: about joules
Example 7

41. CRATE A person pushes a crate along the floor with a constant force of 30 newtons at a constant angle of 30°. Find the work done in joules moving the crate 8 meters.
A joules
B joules
C joules
D joules
Example 7