The area of the rectangle having vertices $P, Q, R, S$ with position vectors $-\hat{i}+\hat{j}+\hat{k}, \hat{i}+\hat{j}+\hat{k}, \hat{i}-\hat{j}+\hat{k}, -\hat{i}-\hat{j}+\hat{k}$ respectively is

If $a$ and $b$ are two unit vectors such that $a+b$ is also a unit vector,then $|a-b|^2=$

If $a$ and $b$ are two non-zero vectors,then the component of $b$ along $a$ is

If $|\vec{a}|=16$ and $|\vec{b}|=4$,then find the value of $\sqrt{|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}}$.

If $a \cdot i = a \cdot (i + j) = a \cdot (i + j + k)$, then $a = $

Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{\alpha}+3 \vec{\beta}$ is collinear with $\vec{\gamma}$ and $\vec{\beta}+2 \vec{\gamma}$ is collinear with $\vec{\alpha}$, then $\vec{\alpha}+3 \vec{\beta}+6 \vec{\gamma}$ is