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

Answer: 3

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