Question

A vector $(a \hat{i}+b \hat{j}+c \hat{k})$ is rotated through a certain angle about the origin in the anti-clockwise direction. If the new vector obtained is $(a-1) \hat{i}+(b-1) \hat{j}+(c-1) \hat{k}$, then find the value of $2(a+b+c)$.

Answer 1

The magnitude of a vector remains unchanged after rotation.
$\Rightarrow|a \hat{i}+b \hat{j}+c \hat{k}|=|(a-1) \hat{i}+(b-1) \hat{j}+(c-1) \hat{k}| $
$\Rightarrow \sqrt{a^{2}+b^{2}+c^{2}}=\sqrt{(a-1)^{2}+(b-1)^{2}+(c-1)^{2}} $
$\Rightarrow a^{2}+b^{2}+c^{2}=a^{2}+b^{2}+c^{2}-2(a+b+c)+3 $
$\Rightarrow 2(a+b+c)=3$