$PQRS$ is a quadrilateral and $PQ=a, QR=b, SP=a-b$. $M$ is the mid-point of $QR$ and $X$ is a point on $SM$ such that $SX=\frac{4}{5}SM$. If $SM=m(4a-b)$ and $SX=n(4a-b)$,then $m+n=$

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$,then $\vec{a} + 2\vec{b} + 6\vec{c} = \dots$

If the position vector of one end of a line segment $AB$ is $2i + 3j - k$ and the position vector of its midpoint is $3(i + j + k)$,then what is the position vector of the other end?

If $|\overline{u}|=2$ and $\overline{u}$ makes angles of $60^{\circ}$ and $120^{\circ}$ with axes $OX$ and $OY$ respectively,then $\overline{u}=$

$A$ vector which is in the direction of $(3, 6, 2)$ and has magnitude $4$ is $.......$

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in the counterclockwise direction in the first quadrant. Then the area of the triangle having vertices $(\alpha, \beta), (0, \beta)$ and $(0,0)$ is equal to