Scalar projection of the line segment joining the points $A(-2,0,3)$ and $B(1,4,2)$ on the line whose direction ratios are $6,-2,3$ is

Let $m$ be the unit vector orthogonal to the vector $\hat{i}-\hat{j}+\hat{k}$ and coplanar with the vectors $2 \hat{i}+\hat{j}$ and $\hat{j}-\hat{k}$. If $a=\hat{i}-\hat{k}$,then the length of the perpendicular from the origin to the plane $r \cdot m=a \cdot m$ is

Vector $\vec{a} + 3\vec{b}$ is perpendicular to $7\vec{a} - 5\vec{b}$ and $\vec{a} - 5\vec{b}$ is perpendicular to $7\vec{a} + 3\vec{b}$. The angle between non-zero vectors $\vec{a}$ and $\vec{b}$ is:

If the vectors $\vec{AB} = p \hat{i} + q \hat{j} + r \hat{k}$, $\vec{AC} = s \hat{i} + 3 \hat{j} + 4 \hat{k}$, and $\vec{CB} = 3 \hat{i} + \hat{j} - 2 \hat{k}$ form a $\triangle ABC$, then the values of $p, q, r$ and $s$ such that the area of that $\triangle ABC$ is $5 \sqrt{6}$ are:

If $|a \times b| = 4$ and $|a \cdot b| = 2$,then $|a|^2 |b|^2 = $

Let $a, b$ and $c$ be three non-coplanar vectors. The vector equation of a line which passes through the point of intersection of two lines,one joining the points $a+2b-5c$ and $-a-2b-3c$,and the other joining the points $-4c$ and $6a-4b+4c$,is