20250702

ベクトルのなす角度

$$ \vec{a} = (0, 1) \ \vec{b} = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $$

ベクトルのなす角度の公式： $$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$$

計算過程

$$ \begin{align} \vec{a} \cdot \vec{b} &= 0 \times \frac{1}{2} + 1 \times \frac{\sqrt{3}}{2} \ &= 0 + \frac{\sqrt{3}}{2} \ &= \frac{\sqrt{3}}{2} \end{align} $$

$$ \begin{align} |\vec{a}| &= \sqrt{0^2 + 1^2} = 1 \ |\vec{b}| &= \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \ &= \sqrt{\frac{1}{4} + \frac{3}{4}} \ &= \sqrt{1} = 1 \end{align} $$

$$ \begin{align} \cos \theta &= \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|} \ &= \frac{\frac{\sqrt{3}}{2}}{1 \times 1} \ &= \frac{\sqrt{3}}{2} \end{align} $$

答え

$$ \theta = \arccos\left(\frac{\sqrt{3}}{2}\right) = 30° = \frac{\pi}{6} \text{ ラジアン} $$