If ( a b) c= a( b c) where a, b and c are any three vectors such that a b0, b c0, then a a…

Question

If $(\vec a	imes\vec b)	imes\vec c=\vec a	imes(\vec b	imes\vec c)$ where $\vec a,\vec b$ and $\vec c$ are any three vectors such that $\vec a\cdot\vec b
e0$, $\vec b\cdot\vec c
e0$, then $\vec a$ and $\vec c$ are:

Accepted Answer

$$(\vec a\cdot\vec c)\vec b-(\vec b\cdot\vec c)\vec a = (\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c \implies (\vec a\cdot\vec b)\vec c=(\vec b\cdot\vec c)\vec a$$

Since $\vec a\cdot\vec b\ne0$ and $\vec b\cdot\vec c\ne0$, this gives $\vec c$ as a scalar multiple of $\vec a$, i.e. $\vec a\parallel\vec c$.

Solutions · 1