Pre-Calculus
Name________________
Assignment: J10
8.4 – 8.5 Quiz Review
Sketch the vector indicated.
1.
3u
2.
–u
3.
-2v
4.
v–u
5.
u+v
u
v
Express the vector with initial point P and terminal point Q in component form.
6.
P(1, 1), Q(9, 7)
7.
P(-1, 3), Q(-6, -1)
Find -2u, u + v, 2v – u, |𝐮|, |𝐯|, and |𝐮 − 𝐯| for the given vectors u and v.
8. 𝐮 = ⟨−2,5⟩, 𝐯 = ⟨2, −8⟩
9. 𝐮 = 𝐢 + 𝐣, 𝐯 = 𝐢 − 𝐣
10. 𝐮 = −2𝐢 + 3𝐣, 𝐯 = 𝐢 − 2𝐣
11. 𝐮 = ⟨−6,6⟩, 𝐯 = ⟨−2, −1⟩
Find the horizontal and vertical components of the vector with the given length and direction.
Write your answer in component form AND in terms of i and j.
12. |𝐯| = 40, 𝜃 =
2𝜋
3
14. |𝐯| = 5, 𝜃 = 225º
13. |𝐯| = √3, 𝜃 = 300º
15. |𝐯| = 2, 𝜃 =
𝜋
2
Find the magnitude and direction of the vector. Write your answer in radians.
2
2
2
2
√
√
16. ⟨− , − ⟩
17. 𝐯 = 𝐢 + 𝐣
18. 𝐮 = −𝐢 − √3𝐣
19. ⟨−4√3, 4⟩
Find 𝐮 ∙ 𝐯 for the given vectors u and v.
20. u = 5i + j, v = 6i – 2j
21. 𝐮 = ⟨3, −5⟩, 𝐯 = ⟨−2, 7⟩
22. 𝐮 = ⟨0,4⟩, 𝐯 = ⟨10, −1⟩
23. u = –3i – 2j, v = 2j
Find the angle between u and v to the nearest degree.
24. 𝐮 = 𝐢 + 𝐣, 𝐯 = 𝐢 − 𝐣
25. 𝐮 = ⟨−6, 6⟩, 𝐯 = ⟨1, −1⟩
26. 𝐮 = ⟨3, 4⟩, 𝐯 = ⟨2, 1⟩
27. 𝐮 = 𝐢 + √3𝐣, 𝐯 = −√3𝐢 + 2𝐣
Determine whether the vectors are orthogonal.
28. 𝐮 = ⟨5, 3⟩, 𝐯 = ⟨−2, 6⟩
29. 𝐮 = 𝐢 − 𝐣, 𝐯 = 𝐢 + 𝐣
Find the indicated quantity, given 𝐮 = 𝟑𝐢 − 𝟐𝐣, 𝐯 = −𝟒𝐢 + 𝐣, 𝐰 = 𝟔𝐣.
30. 𝐮 ∙ 𝐯 + 𝐮 ∙ 𝐰
31. 𝐮 ∙ (𝐯 + 𝐰)
32. (𝐮 + 𝐯) ∙ (𝐮 − 𝐯)
33. (𝐮 ∙ 𝐯)(𝐮 ∙ 𝐰)
Find the work done by the force F in moving an object from P to Q.
34. 𝐅 = 400𝐢 + 50𝐣; P(−1, 1), Q(200, 1)
35. 𝐅 = −4𝐢 + 20𝐣; P(0, 10), Q(5, 25)
36. The force 𝐅 = 5𝐢 − 6𝐣 moves an object 10 feet along the x-axis in the positive direction. Find the work done
if the unit of force is the pound.
37. A lawn mower is pushed a distance of 100 feet along a horizontal path by a constant force of 60 lb. The
handle of the lawn mower is held at an angle of 45º from the horizontal. Find the work done.
Answers:
1. – 5. See Online Key
6. ⟨8, 6⟩
9. -2i – 2j, 2i, i – 3j, √2, √2, 2
7. ⟨−5, −4⟩
8. ⟨4, −10⟩, ⟨0, −3⟩, ⟨6, −21⟩, √29, 2√17, √185
10. 4i – 6j, -i + j, 4i – 7j, √13, √5, √34
11. ⟨12, −12⟩, ⟨−8, 5⟩, ⟨2, −8⟩, 6√2, √5, √65
√3
12. ⟨−20, 20√3⟩, -20i + 20√3j
16. 1,
5𝜋
4
25. 180º
32. -4
𝜋
17. √2, 4
18. 2,
26. 27º
27. 71º
33. 168
3
13. ⟨ 2 , − 2⟩,
4𝜋
3
19. 8,
3
√3
𝐢− 2𝐣
2
5𝜋
6
20. 28
28. Not orthogonal
34. 80,400 ft/lb
35. 280 ft/lb
14. ⟨−
5√2
5√2
5√2
,− 2 ⟩,− 2 𝐢
2
21. -41
22. -4
29. Orthogonal
36. 50 ft/lb
−
5√2
𝐣
2
23. -4
15. ⟨0, 2⟩, 2j
24. 90º
31. – 26
30. -26
37. 3000√2 ft/lb
Answers:
1. – 5. See Online Key
6. ⟨8, 6⟩
9. -2i – 2j, 2i, i – 3j, √2, √2, 2
7. ⟨−5, −4⟩
8. ⟨4, −10⟩, ⟨0, −3⟩, ⟨6, −21⟩, √29, 2√17, √185
10. 4i – 6j, -i + j, 4i – 7j, √13, √5, √34
11. ⟨12, −12⟩, ⟨−8, 5⟩, ⟨2, −8⟩, 6√2, √5, √65
√3
12. ⟨−20, 20√3⟩, -20i + 20√3j
16. 1,
5𝜋
4
25. 180º
32. -4
𝜋
17. √2, 4
18. 2,
26. 27º
27. 71º
33. 168
3
13. ⟨ 2 , − 2⟩,
4𝜋
3
19. 8,
3
√3
𝐢− 2𝐣
2
5𝜋
6
20. 28
28. Not orthogonal
34. 80,400 ft/lb
35. 280 ft/lb
14. ⟨−
5√2
5√2
5√2
,− 2 ⟩,− 2 𝐢
2
21. -41
22. -4
29. Orthogonal
36. 50 ft/lb
−
5√2
𝐣
2
23. -4
15. ⟨0, 2⟩, 2j
24. 90º
31. – 26
30. -26
37. 3000√2 ft/lb
Answers:
1. – 5. See Online Key
6. ⟨8, 6⟩
9. -2i – 2j, 2i, i – 3j, √2, √2, 2
7. ⟨−5, −4⟩
8. ⟨4, −10⟩, ⟨0, −3⟩, ⟨6, −21⟩, √29, 2√17, √185
10. 4i – 6j, -i + j, 4i – 7j, √13, √5, √34
11. ⟨12, −12⟩, ⟨−8, 5⟩, ⟨2, −8⟩, 6√2, √5, √65
12. ⟨−20, 20√3⟩, -20i + 20√3j
16. 1,
5𝜋
4
25. 180º
32. -4
𝜋
17. √2, 4
18. 2,
26. 27º
27. 71º
33. 168
√3
3
13. ⟨ 2 , − 2⟩,
4𝜋
3
19. 8,
3
√3
𝐢− 2𝐣
2
5𝜋
6
20. 28
28. Not orthogonal
34. 80,400 ft/lb
35. 280 ft/lb
14. ⟨−
5√2
5√2
5√2
,− 2 ⟩,− 2 𝐢
2
21. -41
22. -4
29. Orthogonal
36. 50 ft/lb
−
5√2
𝐣
2
23. -4
30. -26
37. 3000√2 ft/lb
15. ⟨0, 2⟩, 2j
24. 90º
31. – 26