Discussion
If \(|\vec{a}| = 8, |\vec{b}| = 5\), and \(|\vec{a} - \vec{b}| = 7\) then the angle between \(\vec{a}\) and \(\vec{b}\) is equal to
Answer: E
The angle is \(\frac{\pi}{3}\).
Expand \(|\vec{a} - \vec{b}|^{2}\):
\[ \begin{aligned} 49 &= 64 + 25 - 2(8)(5)\cos\theta \\ 80\cos\theta &= 40 \;\Rightarrow\; \cos\theta = \tfrac{1}{2} \end{aligned} \]
so \(\theta = \frac{\pi}{3}\). \(\frac{2\pi}{3}\) is what \(|\vec{a} + \vec{b}| = 7\) would give, the sign trap; \(\frac{\pi}{4}\), \(\frac{3\pi}{4}\) and \(\frac{\pi}{6}\) do not satisfy \(\cos\theta = \frac{1}{2}\).
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