If the points $P$ and $Q$ are respectively the circumcentre and the orthocentre of a $\triangle ABC$,then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ is equal to

Two adjacent sides of a parallelogram $PQRS$ are given by $\vec{PQ} = \hat{i} + \hat{k}$ and $\vec{PS} = \hat{i} - \hat{j}$. If the side $PS$ is rotated about the point $P$ by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side $PQ$, then $\sin^2(\frac{5\alpha}{2}) - \sin^2(\frac{\alpha}{2})$ is equal to:

If the vectors $a\hat{i} + 2\hat{j} + 3\hat{k}$ and $-\hat{i} + 5\hat{j} + a\hat{k}$ are perpendicular to each other,then $a = $

The locus of a point equidistant from two given points $a$ and $b$ is given by

If $(a \times b)^{2} + (a \cdot b)^{2} = 144$ and $|a| = 4$,then $|b|$ is equal to

The vector $b = 3j + 4k$ is to be written as the sum of a vector $b_1$ parallel to $a = i + j$ and a vector $b_2$ perpendicular to $a$. Then $b_1 = $