Let a, b, c be unit vectors, b, c non-parallel.

Question

Let $\vec a,\vec b,\vec c$ be unit vectors, $\vec b,\vec c$ non-parallel. If $\alpha,\beta$ are the angles $\vec a$ makes with $\vec b,\vec c$ respectively, and $\vec a	imes(\vec b	imes\vec c)=\dfrac12\vec b$, then $|\alpha-\beta|$ is:

Accepted Answer

$\vec a\times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c = \dfrac12\vec b$. Since $\vec b,\vec c$ are non-parallel (independent): $\vec a\cdot\vec c=\dfrac12$, $\vec a\cdot\vec b=0$.

$$\cos\beta=\frac12 \implies \beta=60^{\circ},\qquad \cos\alpha=0 \implies \alpha=90^{\circ}$$

$$|\alpha-\beta| = 90^{\circ}-60^{\circ}=30^{\circ}$$

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