44 $\sqrt{75}$ , $(-25, 5, 35)$ )
45 $(1, -13, -7)$ , $(-6, 21, -12)$
48 (a) $(8, 1, 6)$ , $(4, 1, 3)$      (b) $(-\frac{3}{5}, 0, \frac{4}{5})$      (c) $\frac{5}{2}$
51 $(48, 72, 0)/\sqrt{14}$
52 $(-8, -32, -4)/\sqrt{21}$
53 (a) $(0, 1, -1)$    $(-2, 1, -1)$    $(-2, 1, 0)$
     (b) $(0, 0, 0)$    $(0, 0, -2)$    $(-3, 3, -1)$
     (c) $(-3, 3, -3)$
54 $\pm(-3, 5, 11)/\sqrt{155}$    0.9968
55 Distance = 1.92
56 $m\omega = eB$
57 15
59 8
61 (a) $(-5, 3, -7)/\sqrt{83}$    (b) $(0, 1, -4)/\sqrt{17}$
62 $\frac{7}{3}(1, 1, 1)$    $\frac{1}{3}(2, -13, 11)$
63 $\begin{vmatrix} u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{vmatrix}$
64 $\alpha = -1/F^2$
65 (a) $(\mathbf{c} \cdot \mathbf{a})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{c} \cdot \mathbf{b})(\mathbf{d} \cdot \mathbf{a})$    (c) $-(\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \times \mathbf{c})$
66 (a) $(3, 3, 3)$    (b) $(1 + s, 2 + s, 3 + s)$
     (c) $x - 1 = y - 2 = z - 3$
67 yes, no, no, yes $\mathbf{r} = (2 + s, 1 - s, 1 - s)$
69 $(3, 4, 0)$ , $43.5^\circ = 0.759 \text{ rad}$
71 $\mathbf{r} = (2 - t, 2t, -1 + 4t)$
     $2 - x = \frac{1}{2}y = \frac{1}{4}(1 + z)$ , 无交点
72 $\sqrt{35}$
73 $\mathbf{r} \cdot (0, -1, 1) = 1 \quad -y + z = 1$
74 $\mathbf{r} \cdot (1, -1, 1) = 2$
75 $\mathbf{r} \cdot [\mathbf{b} \times (\mathbf{c} - \mathbf{a})] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$
76 $\mathbf{r} = (0, -5, 10) + \lambda(1, 2, -3)$
77 $\mathbf{r} = (1, 2, 4) + t(1, 1, 2)$    $(-\frac{5}{2}, -\frac{3}{2}, -3)$
78 $79.0^\circ = 1.38 \text{ rad}$
79 (a) $\mathbf{r} \cdot (2, 3, 6) = -28$    (b) 5
80 $\mathbf{r} = (1 + 2t, -1 + 4t, 3 - 4t)$ , $41.8^\circ = 0.729 \text{ rad}$
81 $\mathbf{r} \cdot (1, -5, 3) = 28$
82 $\mathbf{r} = (-1 + 14t, t, 1 - 8t)$
     $\frac{2}{3}\sqrt{29}$ , $\mathbf{r} \cdot (-18, 36, -27) = -9$

4.6 复习题

1 (a) $\sqrt{93}$    (b) $(17, -3, -10)/\sqrt{398}$
     (c) $85.8^\circ = 1.50 \text{ rad}$ , $47.0^\circ = 0.820 \text{ rad}$
     (d) $(2, 13, -13)/6$
2 (a) $(3, 4, 5)$    (b) $\sqrt{35}$    (c) $34/3$
3 (a) $(1, 2, 0)$ , $(2, 1, 1)$    (b) $\sqrt{5}$
     (c) 1      (d) $112.2^\circ = 1.96 \text{ rad}$
4 (a) $-4$    (b) 1 or $-4$
5 $(\frac{2}{3}, \frac{1}{3}, \frac{2}{3})$ , $(-\frac{3}{5}, 0, \frac{4}{5})$ ; $(1, 5, 22)$
8 $(1, 1, 1)$ , $(-5, -11, 1)$
9 $\mathbf{E} = e(0.550, 0.282, 0.282)$
10 (a) $2x + 3y + 6z + 28 = 0$    (b) 5
11 (a) $2x + 3y - z = 10$ ; $10/\sqrt{14}$    (b) $\sqrt{3/2}$
12 P(2, 4, 4), Q(1, 2, 3)
     $(-2, -4, 0)$
13 (a) 0    (b) 15(1, 1, -2)
15 $(-90, -36, 12)$ , $85.3^\circ$ or 1.49 rad
16 $(11, -12, 5)$ ; $76.8^\circ$ ; $(-11, 12, -5)/\sqrt{290}$
     (a) $-11x + 12y - 5z = 8$
     (b) $-11x + 12y - 5z = -4$    (c) $12/\sqrt{290}$
17 $\mathbf{r} = (-3, 0, 1) + \lambda(8, -8, -8) + \mu(5, 1, -3)$
     $\mathbf{r} \cdot (-1, 2, -1) = -6$
18 $x + 2y - 2z = -1$
19 (a) $(0, 0, 1)$ , $(1, -1, 0)$ , $(0, 1, -1)$
     (b) 1    (c) 3, -3, 2    (d) 3, 2, 1
20 $\mathbf{r} = (2 - t, 3 - 3t, 2t)$
     (a) $\sqrt{61/14}$    (b) $(0, -3, 4)$    (c) $(19, 15, 18)/14$
21 $\alpha = \mathbf{r} \cdot \mathbf{a}' \quad \beta = \mathbf{r} \cdot \mathbf{b}' \quad \gamma = \mathbf{r} \cdot \mathbf{c}'$
22 Taking $i$ 沿 OA 和 $j$ 沿 OB 方向，则
$\mathbf{F} = \omega^2(1.4, 1.65)$ and $\text{OC} = (-1.4, -1.65) \text{ m.}$

CHAPTER 5

Exercises

1 (a) 请提供需要翻译的 Markdown 内容。    (b) $\begin{bmatrix} 1 \ 3 \ 1 \end{bmatrix}$    (c) 请提供需要翻译的 Markdown 文本。