Calculus 2nd Edition Rogawski

672 C H A P T E R 13 VECTOR GEOMETRY
v
w
(A) (B) (C)
FIGURE 21
16. Sketch v + w and v − w for the vectors in Figure 22.
v
FIGURE 22
17. Sketch 2v, −w, v + w, and 2v − w for the vectors in Figure 23.
5
4
3
2
1
y
v = ⟨2, 3⟩
1 2 3 4 5 6
FIGURE 23
w
w = ⟨4, 1⟩
18. Sketch v = ⟨1, 3⟩, w = ⟨2, −2⟩, v + w, v − w.
19. Sketch v = ⟨0, 2⟩, w = ⟨−2, 4⟩, 3v + w, 2v − 2w.
20. Sketch v = ⟨−2, 1⟩, w = ⟨2, 2⟩, v + 2w, v − 2w.
21. Sketch the vector v such that v + v 1 + v 2 = 0 for v 1 and v 2 in
Figure 24(A).
22. Sketch the vector sum v = v 1 + v 2 + v 3 + v 4 in Figure 24(B).
−3
v 2
3
1
y
(A)
1
v 1
x
FIGURE 24
x
y
v 1
v 3
v 4
v 2
23. Let v = −→ PQ, where P = (−2, 5), Q = (1, −2). Which of the following
vectors with the given tails and heads are equivalent to v?
(a) (−3, 3), (0, 4) (b) (0, 0), (3, −7)
(c) (−1, 2), (2, −5) (d) (4, −5), (1, 4)
24. Which of the following vectors are parallel to v = ⟨6, 9⟩ and which
point in the same direction?
(a) ⟨12, 18⟩ (b) ⟨3, 2⟩ (c) ⟨2, 3⟩
(d) ⟨−6, −9⟩ (e) ⟨−24, −27⟩ (f) ⟨−24, −36⟩
(B)
x
In Exercises 25–28, sketch the vectors −→ AB and −→ PQ, and determine
whether they are equivalent.
25. A = (1, 1), B = (3, 7), P = (4, −1), Q = (6, 5)
26. A = (1, 4), B = (−6, 3), P = (1, 4), Q = (6, 3)
27. A = (−3, 2), B = (0, 0), P = (0, 0), Q = (3, −2)
28. A = (5, 8), B = (1, 8), P = (1, 8), Q = (−3, 8)
In Exercises 29–32, are −→ AB and −→ PQparallel? And if so, do they point
in the same direction?
29. A = (1, 1), B = (3, 4), P = (1, 1), Q = (7, 10)
30. A = (−3, 2), B = (0, 0), P = (0, 0), Q = (3, 2)
31. A = (2, 2), B = (−6, 3), P = (9, 5), Q = (17, 4)
32. A = (5, 8), B = (2, 2), P = (2, 2), Q = (−3, 8)
In Exercises 33–36, let R = (−2, 7). Calculate the following.
33. The length of −→ OR
34. The components of u = −→ PR, where P = (1, 2)
35. The point P such that −→ PR has components ⟨−2, 7⟩
36. The point Q such that −→ RQ has components ⟨8, −3⟩
In Exercises 37–42, find the given vector.
37. Unit vector e v where v = ⟨3, 4⟩
38. Unit vector e w where w = ⟨24, 7⟩
39. Vector of length 4 in the direction of u = ⟨−1, −1⟩
40. Unit vector in the direction opposite to v = ⟨−2, 4⟩
41. Unit vector e making an angle of 4π 7
with the x-axis
42. Vector v of length 2 making an angle of 30 ◦ with the x-axis
43. Find all scalars λ such that λ ⟨2, 3⟩ has length 1.
44. Find a vector v satisfying 3v + ⟨5, 20⟩ = ⟨11, 17⟩.
45. What are the coordinates of the point P in the parallelogram in
Figure 25(A)?
46. What are the coordinates a and b in the parallelogram in Figure
25(B)?
y
P
(2, 2)
(A)
(7, 8)
(5, 4)
x
(−3, 2)
FIGURE 25
(−1, b)
y
(B)
(a, 1)
(2, 3)
47. Let v = −→ AB and w = −→ AC, where A, B, C are three distinct points
in the plane. Match (a)–(d) with (i)–(iv). (Hint: Draw a picture.)
x