第12章 12.4

(1)

$a \cdot b = 2 \cdot 1 + 0 \cdot 1 = 2$

(2)

$b \cdot c = 1 \cdot 1 + 1 \cdot \sqrt{3} = 1 + \sqrt{3}$

(3)

$\cos\theta = \frac{a \cdot b}{|a||b|} = \frac{2}{\sqrt{2^2+0^2}\sqrt{1^2+1^2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \qquad \theta = \frac{\pi}{4}$

(4)

$\cos\theta = \frac{a \cdot c}{|a||c|} = \frac{2}{\sqrt{2^2+0^2}\sqrt{1^2+\sqrt{3}^2}} = \frac{2}{2 \cdot 2} = \frac{1}{2} \qquad \theta = \frac{\pi}{3}$

(5)

求めるベクトルを $d(a,b)$ とすると

$b \perp d \quad \text{から} \quad b \cdot d = 0 \qquad ①$

$\text{大きさが2\text{なので}} \quad 2 = |d| \qquad ②$

$①\text{から} \quad 1 \cdot a + 1 \cdot b = 0 \qquad a = -b \qquad ③$

$②\text{から} \quad 2 = \sqrt{a^2+b^2} \qquad 4 = a^2+b^2 \qquad ③\text{を代入して} \qquad 4 = 2a^2 \qquad a = \pm\sqrt{2} \qquad \text{これから} \qquad b = \mp\sqrt{2}$

$\text{よって} \quad (\pm\sqrt{2}, \mp\sqrt{2}) \quad \text{複合同順}$