If a vector $\vec a$ is expressed as the sum of two vectors $\vec α$ and $\vec β$ along and perpendicular to a given vector $\vec b$, then $\vec β$ is equal to
Answer & explanation
Correct answer: option 2
We have,
$\vec α$ = Projection vector of $\vec a$ on $\vec b$ $⇒\vec α=\frac{(\vec a.\vec b)}{|\vec b|^2}\vec b$
Clearly, $\vec α+\vec β=\vec a$
$⇒\vec β=\vec a-\frac{(\vec a.\vec b)}{|\vec b|^2}\vec b$
$⇒\vec β=\frac{|\vec b|^2\vec a-(\vec a.\vec b)\vec b}{|\vec b|^2}=\frac{\vec b×(\vec a×\vec b)}{|\vec b|^2}$
