So, the question is as follows :
If $p,q,r$ are mutually perpendicular vectors such that $|p| = |q| = |r| \neq 0$.
then, is $(p \cdot x)p +(q \cdot x)q +(r \cdot x)r =|p|^2 x$? where $x$ is not a null vector.
my approach:
since $p,q,r$ are mutually perpendicular vectors then we can take them along the $x,y,z$ coordinate axes and it is obvious that the result is true. But I do not know if this can be done without loss in generality. Is the result true? All help is greatly appreciated.