4 (a) 圆心 $(-\frac{1}{3}, \frac{4}{3})$ , radius $\frac{2}{3}\sqrt{2}$

$$(b) \operatorname{Re}\left(\frac{1}{z-2}\right) = -\frac{1}{2}$$

6 Centre $(R_2, \frac{1}{2}\omega L)$ , radius $\frac{1}{2}\omega L$

7 (a) $32\cos^6\theta - 48\cos^4\theta + 18\cos^2\theta - 1$

13 $419.8 - j238.8, 0.5928 \times 10^{-3} + j1.0518 \times 10^{-3}$

14 $1 + j3, \frac{1}{5}(3 + j11), \frac{1}{5}(7 + j11)$

15 $\operatorname{Mod} \frac{25}{13}, \arg = -154^\circ 17' = -2.6927 \text{ rad}$

16 (a) $0.22 \pm j0.49$ (b) $1.44 + j1.57$ (c) $10.48 + j19.74$
(d) $0.80 + j0.46$ (e) $1.09 + j0.83$

$$19 \quad \theta = \tan^{-1}\left[\frac{2R_0X - 2RX_0}{R^2 + X^2 - R_0^2 - X_0^2}\right]$$

20 $4.46 - j2.06$

21 (a) $0.7974 + j0.3685$ (b) $r = 0.8784$
$\theta = 24^\circ 49' = 0.4329 \text{ rad}, 1.098$

23 (a) $-0.04 + j0.28$ (b) $\pm(0.35 + j0.40)$
(c) $0.92 + j0.27$ (d) $-1.26 + j1.71$
(e) $-0.04 + j0.28$

24 $1 \angle 18^\circ 26', 1 \angle 108^\circ 26', 1 \angle 198^\circ 26', 1 \angle 288^\circ 26'$

25 $2^{1/6}e^{j(1/9 + k/3)\pi}, k = 0, \dots, 5$

27 $-(\omega u + \omega^2 v), -(\omega^2 u + \omega v); \frac{1}{4}r^2 \leq -\frac{1}{27}q^3$

28 $1 - j2, 2\sqrt{5}$

31 Circle $u^2 + v^2 - 12u + 16v = 0$
Centre $(6, -8)$ , radius 10

32 $v + 3u = 5$

33 Circle $u^2 + v^2 - \frac{5}{2}u + 1 = 0$ ; Centre $(\frac{5}{4}, 0)$ , radius $\frac{3}{4}$ ;
映射到圆外区域

CHAPTER 4

Exercises

2 183.3 km, 270 km

4 $60^\circ$ or $-60^\circ$ 到正的 $z$ axis

5 $\frac{1}{3}a + \frac{2}{3}b$

7 $\vec{OC} = 2a + b, \vec{OD} = 2a + 2b, \vec{OE} = a + 2b$

8 $20.62 \text{ ms}^{-1}, 14.05^\circ$

9 $8\sqrt{2}$ 来自西北方向的每小时千米

10 70.71 N

11 (a) $(3, 3, 1)$ (b) $(2, 4, \frac{5}{2})$ (c) $(0, 0, 1)$
(d) $\sqrt{2}$ (e) 3 (f) $\sqrt{3}$
(g) $(\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{2}}, 0)$ (h) $(\frac{2}{3}, \frac{2}{3}, \frac{1}{3})$

12 $\vec{PQ} = (4, -5, 11), |\vec{PQ}| = 9\sqrt{2}$
方向余弦 $4/(9\sqrt{2}), -5/(9\sqrt{2}), 11/(9\sqrt{2})$

13 $\sqrt{134} \text{ N}, (7, 2, 9)/\sqrt{134}$

14 $\alpha = 4 \quad \beta = 1 \quad \gamma = 2$

15 $\sqrt{21} \quad \sqrt{17} \quad \sqrt{38}$

16 $(1, 1, -1.414)$

17 (a) $\vec{AB} = (6, 0, -1), \vec{AC} = (1, 2, -1)$ (b) 7
(c) $(\frac{3}{7}, \frac{6}{7}, \frac{1}{7})$ (d) $\sqrt{37}, \sqrt{6}$ (e) $\frac{1}{2}(7, 2, -2)$

18 $(1, 4), (4, 0)$

19 $\vec{PQ} = \vec{QR} = (1, 5, -3)$ and $PQ : QR = 1 : 1$

20 distance = $13/5, t = 1/5$

22 $1 - j2$ length = $\sqrt{20}$

23 $\frac{1}{2} - j\frac{1}{2}$

25 $\theta = \sin^{-1}\left(\frac{W_1^2 + W_2^2 - W_3^2}{2W_1W_2}\right)$

$\phi = \sin^{-1}\left(\frac{W_2^2 + W_3^2 - W_1^2}{2W_2W_3}\right)$

26 $F = (-940, 124, -31) \text{ N}$
(7.93 m, -1.04 m, 0 m), $T = 1342 \text{ N}$

27 (a) 14 (b) 6 (c) $(2, 1, 6)/\sqrt{41}$
(d) $(12, 0, -6)/\sqrt{5}$ (e) $-24$ (f) $(12, 4, 8)$

28 (a) $98.0^\circ = 1.711 \text{ rad}$ (b) $64.8^\circ = 1.130 \text{ rad}$
(c) $\frac{14}{5}$ (d) 3 or $-4$

29 $1, -1, 2$

30 $\sqrt{45}, \sqrt{55}, 27.8^\circ$

31 4 units

32 $\frac{5}{14}$

33 $\sqrt{5}/2$

38 $r^2 - (r \cdot \hat{a})^2 = R^2$

39 $\sqrt{3}, 70.5^\circ$ or $1.23 \text{ rad}$

40 $|X| \leq 2.98 \text{ m}$

41 (a) $(3, -2, -1)$ (b) $(-1, 1, 0)$ (c) $(5, -4, -2)$
(d) $-1$ (e) 1 (f) $(2, 2, 1)$

42 (a) $(-5, -3, 1), (-10, -6, 2)$
(b) $(5, 3, -1), (10, 6, -2)$
(c) $a$ and $c$ 是平行的