第12章 12.18

(1)

$\vec{a} = \vec{e_1} + 3\vec{e_2}$

$\vec{b} = 2\vec{e_1} + t\vec{e_2}$

$\vec{a} \cdot \vec{b} = 0$

$(\vec{e_1} + 3\vec{e_2})(2\vec{e_1} + t\vec{e_2}) = 0$

$2|\vec{e_1}|^2 + 3t|\vec{e_2}|^2 = 0$

$2 + 3t = 0$

$t = -\frac{2}{3}$

(2)

$\vec{a} = (1,1) \qquad \vec{b} = (1,-\sqrt{3})$

$(t\vec{a} + \vec{b}) \cdot \vec{a} = 0$

$(1+t, -\sqrt{3}+t) \cdot (1,1) = 0$

$1 + t - \sqrt{3} + t = 0$

$t = \frac{\sqrt{3} - 1}{2}$