Question: 4. Compute 4w − u.
5. Compute −u + v + w.
6. Compute 3u − 2v + 5w.
In Exercises 7–10, express the given vector equation as a system of
linear equations.
7. x1 [3
2] + x2 [-1
5] = [8
13]
8. x1 [-1
6
-4] + x2 [9
-5
0] = [-7
-11
3]
9. x1 [-6
5] + x2 [5
-3] + x3 [0
2] = [4
16]
10. x1 [2
7
8
3] + x2 [0
2
4
2] + x3 [5
1
6
1] + x4 [4
5
7
0] = [0
4
3
5]
In Exercises 11–14, express the given system of linear equations as
a vector equation.
11. 2x1 + 8x2 − 4x3 = −10
−x1 − 3x2 + 5x3 = 4
14. −5x1 + 9x2 = 13
3x1 − 5x2 = −9
x1 − 2x2 = −2
In Exercises 15–18, the general solution to a linear system is given.
Express this as a linear combination of vectors.
15. x1 = −4 + 3s1
x2 = s1
16. x1 = 7 − 2s1
x2 = −3
x3 = s1
17. x1 = 4 + 6s1 − 5s2
x2 = s2
x3 = −9 + 3s1
x4 = s1
18. x1 = 1 − 7s1 + 14s2 − s3
x2 = s3
x3 = s2
x4 = −12 + s1
x5 = s1
In Exercises 19–22, find three different vectors that are a linear
combination of the given vectors.
19. u = [3
−2], v = [-1
-4]
20. u = [7
1
-13], v = [5
-3
2]
21. u = [-4
0
-3], v = [-2
-1
5], w = [9
6
11]
22. u = [1
8
2
2], v = [4
-2
5
-5], w = [9
9
0
1]
In Exercises 23–26, a vector equation is given with some unknown
entries. Find the unknowns.
23. -3 [a
3] + 4 [-1
b] = [-10
19]
24. 4 [4
a] + 3 [-3
5] - 2 [b
8] = [-1
7]
25. - [ -1
a
2] + 2 [3
-2
b] = [c
-7
8]
26. - [a
4
-2
-1] + 2 [5
1
b
3] - [2
c
-3
-6] = [11
-4
3
d]
In Exercises 27–30, determine if b is a linear combination of the
other vectors. If so, write b as a linear combination.
27. a1 = [-2
5], a2 = [7
-3], b = [8
9]
28. a1 = [2
-3
1], a2 = [0
3
-3], b = [1
-5
-2]
29. a1 = [2
-3
1], a2 = [0
3
-3], b = [6
3
-9]
30. a1 = [2
-3
1], a2 = [0
3
-3], a3 = [-2
-1
3], b = [2
-4
5]
Exercises 31–32 refer to Vigoro and Parker’s fertilizers described
at the start of the section. Determine the total amount of nitrogen,
phosphoric acid, and potash in the given mixture.
31. 200 pounds of Vigoro, 100 pounds of Parker’s.
32. 400 pounds of Vigoro, 700 pounds of Parker’s.
33. Compute u − v.
34. Compute −5u.
35. Compute w + 3v.
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SECTION 2.1 Vectors 55
36. 159 pounds of nitrogen, 109 pounds of phosphoric acid, and
36 pounds of potash.
Exercises 37–40 refer to Vigoro and Parker’s fertilizers described
at the start of the section. Show that it is not possible to combine
Vigoro and Parker’s to obtain the specified mixture of nitrogen,
phosphoric acid, and potash.
37. 148 pounds of nitrogen, 131 pounds of phosphoric acid, and
40 pounds of potash.
38. 100 pounds of nitrogen, 120 pounds of phosphoric acid, and
40 pounds of potash.
39. 25 pounds of nitrogen, 72 pounds of phosphoric acid, and 14
pounds of potash.
40. 301 pounds of nitrogen, 8 pounds of phosphoric acid, and 38
pounds of potash.
One 8.3 ounce can of Red Bull contains energy in two forms: 27
grams of sugar and 80 milligrams of caffeine. One 23.5 ounce
can of Jolt Cola contains 94 grams of sugar and 280 milligrams
of caffeine. In Exercises 41–44, determine the number of cans of
each drink that when combined will contain the specified nerve-
jangling combination of sugar and caffeine.
41. 148 grams sugar, 440 milligrams caffeine.
42. 309 grams sugar, 920 milligrams caffeine.
43. 242 grams sugar, 720 milligrams caffeine.
44. 457 grams sugar, 1360 milligrams caffeine.
One serving of Lucky Charms contains 10% of the percent daily
values (PDV) for calcium, 25% of the PDV for iron, and 25% of the
PDV for zinc. One serving of Raisin Bran contains 2% of the PDV for calcium, 25% of the PDV for iron, and 10% of the PDV for zinc. In Exercises 45–48, determine the number of servings of each cereal required to get the given mix of nutrients.
45. 40% of the PDV for calcium, 200% of the PDV for iron, and
125% of the PDV for zinc.
46. 34% of the PDV for calcium, 125% of the PDV for iron, and
95% of the PDV for zinc.
47. 26% of the PDV for calcium, 125% of the PDV for iron, and
80% of the PDV for zinc.
48. 38% of the PDV for calcium, 175% of the PDV for iron, and
115% of the PDV for zinc.
49. An electronics company has two production facilities, A and B. During an average week, facility A produces 2000 computer monitors and 8000 flat panel televisions, and facility B produces 3000 computer monitors and 10,000 flat panel televisions.
1. **Problem:** Compute $4w - u$ given vectors $u$, $v$, and $w$.
2. **Formula:** Vector subtraction and scalar multiplication are performed component-wise:
$$a\mathbf{x} = (a x_1, a x_2, \ldots), \quad \mathbf{x} - \mathbf{y} = (x_1 - y_1, x_2 - y_2, \ldots)$$
3. **Intermediate work:**
- Multiply each component of $w$ by 4.
- Subtract each component of $u$ from the result.
4. **Explanation:** Multiply vector $w$ by 4, then subtract vector $u$ component-wise.
5. **Final answer:** $4w - u = (4w_1 - u_1, 4w_2 - u_2, \ldots)$.
Since the user provided multiple problems, only the first problem is solved here.
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