Discussion
Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\), \(|\vec{c}| = 4\) and \(\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0\) If the angle between \(\vec{b}\) and \(\vec{c}\) is \(\dfrac{\pi}{3}\) , then \(\vec{a}\) is equal to
Answer: D
\(\vec{a}\) equals \(\pm\dfrac{1}{3\sqrt{3}}(\vec{b} \times \vec{c})\).
\(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\), so it is parallel to \(\vec{b} \times \vec{c}\): \(\vec{a} = \lambda(\vec{b} \times \vec{c})\). Compare lengths:
\[ \begin{aligned} |\vec{b} \times \vec{c}| &= 3 \times 4 \times \sin\tfrac{\pi}{3} = 6\sqrt{3} \\ |\lambda| \times 6\sqrt{3} &= 2 \Rightarrow |\lambda| = \frac{1}{3\sqrt{3}} \end{aligned} \]
The sign is not fixed, since either direction along the perpendicular works. Using \(\cos\frac{\pi}{3}\) instead of the sine, or dropping a factor, gives the other coefficients.
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