Discussion
Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors, and θ be the angle between them.If \(\vec{a} - \vec{b}\) is a unit vector,then θ is equal to
Answer: A
\(\theta\) is \(\frac{\pi}{3}\).
Square the length of \(\vec{a} - \vec{b}\), using \(|\vec{a}| = |\vec{b}| = 1\):
\[ \begin{aligned} |\vec{a} - \vec{b}|^{2} &= 1 + 1 - 2\cos\theta = 1 \\ \cos\theta &= \frac{1}{2} \end{aligned} \]
So \(\theta = \frac{\pi}{3}\): the two unit vectors and their difference form an equilateral triangle. \(\frac{2\pi}{3}\) is the answer when the sum \(\vec{a} + \vec{b}\) is a unit vector, and \(\frac{\pi}{2}\) would make the difference \(\sqrt{2}\) long.
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