If a, b, c are non-zero vectors and n is a unit vector perpendicular to c such that a= b- …

Question

If $\vec a,\vec b,\vec c$ are non-zero vectors and $\hat n$ is a unit vector perpendicular to $\vec c$ such that $\vec a=\alpha\vec b-\hat n$ ($\alpha
e0$), and $\vec b\cdot\vec c=12$, then $\vec c	imes(\vec a	imes\vec b)$ [magnitude] is:

Accepted Answer

$\vec c\times(\vec a\times\vec b) = (\vec c\cdot\vec b)\vec a - (\vec c\cdot\vec a)\vec b$. Since $\hat n\perp\vec c$: $\vec c\cdot\vec a=\alpha(\vec c\cdot\vec b)$, so
$$\vec c\times(\vec a\times\vec b) = (\vec c\cdot\vec b)\vec a-\alpha(\vec c\cdot\vec b)\vec b = (\vec c\cdot\vec b)(\vec a-\alpha\vec b) = (\vec c\cdot\vec b)(-\hat n)$$

$$|\vec c\times(\vec a\times\vec b)| = |\vec c\cdot\vec b|\,|\hat n| = 12\times1 = 12$$

Solutions · 1