Discussion
If \(|\vec{a} + \vec{b}| = \frac{\sqrt{14}}{2}\) where \(\vec{a}\) and \(\vec{b}\) are unit vectors, then the value of \(|\vec{a} + \vec{b}|^{2} - |\vec{a} - \vec{b}|^{2}\) is equal to
Answer: A
The value is 3.
For any vectors, \(|\vec{a} + \vec{b}|^{2} - |\vec{a} - \vec{b}|^{2} = 4\,\vec{a}\cdot\vec{b}\). Find \(\vec{a}\cdot\vec{b}\) from the given length, using \(|\vec{a}| = |\vec{b}| = 1\):
\[ \begin{aligned} |\vec{a} + \vec{b}|^{2} &= 2 + 2\,\vec{a}\cdot\vec{b} = \frac{14}{4} \\ \vec{a}\cdot\vec{b} &= \frac{3}{4} \\ 4\,\vec{a}\cdot\vec{b} &= 3 \end{aligned} \]
Check: \(|\vec{a} - \vec{b}|^{2} = 2 - \frac{3}{2} = \frac{1}{2}\), and \(\frac{7}{2} - \frac{1}{2} = 3\). 7 and \(\sqrt{7}\) come from mishandling \(\left(\frac{\sqrt{14}}{2}\right)^{2}\).
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