The unit vector parallel to the resultant vector of $2i + 4j - 5k$ and $i + 2j + 3k$ is

The vectors $3i + j - 5k$ and $ai + bj - 15k$ are collinear if $....$

If $a, b, c$ are three linearly independent vectors and there exists a non-zero scalar triad $(l, m, n)$ such that $l(3a + 2b + c) + m(2a + 2b + 3c) + n(a + 2b + 5c) = 0$, then:

$\vec{c}$ is a unit vector in the direction of the sum of vectors $\vec{a}$ and $\vec{b}$. Where,$\vec{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}$,then $|\vec{c}| = $ . . . . . . .

If in the given figure $\overrightarrow{OA} = \vec{a}$,$\overrightarrow{OB} = \vec{b}$ and $AP : PB = m : n$,then $\overrightarrow{OP} = $

$PQRS$ 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 $m+n=$