The points represented by $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are coplanar and $(\sin A)\vec{a} + (2\sin 2B)\vec{b} + (3\sin 3C)\vec{c} - 4\vec{d} = \vec{0}$. Then the least value of $\frac{21}{8}(\sin^2 A + \sin^2 2B + \sin^2 3C)$ is:

If $a$ and $b$ are mutually perpendicular vectors,then $(a + b)^2 = $

The value of $m \in R$,when the angle between the vectors $\bar{p} = m \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\bar{q} = \hat{i} + 2 \hat{j} + 2m \hat{k}$ is obtuse,is

Let $\vec{a}=\hat{i}+2 \hat{j}-2 \hat{k}$ and $\vec{b}=2 \hat{i}-\hat{j}-2 \hat{k}$ be two vectors. If the orthogonal projection vector of $\vec{a}$ on $\vec{b}$ is $\vec{x}$ and the orthogonal projection vector of $\vec{b}$ on $\vec{a}$ is $\vec{y}$, then find $|\vec{x}-\vec{y}|$.

If $|\vec{a}|=\sqrt{26}$,$|\vec{b}|=7$ and $|\vec{a} \times \vec{b}|=35$,then $\vec{a} \cdot \vec{b}$ is-

If $\vec{i}+\vec{j}-\vec{k}, -\vec{i}+2\vec{j}+\vec{k}, \vec{j}+2\vec{k}, 2\vec{i}-\vec{j}+2\vec{k}$ are the position vectors of four points $A, B, C, D$ respectively,then the shortest distance between the lines $AB$ and $CD$ is