If the position vectors of three points $A, B, C$ are $\hat{i}+2\hat{j}+\hat{k}$,$2\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+\hat{j}+2\hat{k}$ respectively,then the perpendicular distance of the point $C$ from the line $AB$ is

If $a \perp b$ and $(a+b) \perp (a+mb)$,then $m$ is equal to

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (2 \hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + 3 \hat{k}) = 1$,then $\overrightarrow{a}$ is equal to :

Let $\bar{a} = \bar{i} + 2\bar{j} + 3\bar{k}$,$\bar{b} = 2\bar{i} - 3\bar{j} + \bar{k}$,and $\bar{c} = 3\bar{i} + \bar{j} - 2\bar{k}$ be three vectors. If $\bar{r}$ is a vector such that $\bar{r} \cdot \bar{a} = 0$,$\bar{r} \cdot \bar{b} = -2$,and $\bar{r} \cdot \bar{c} = 6$,then find the value of $\bar{r} \cdot (3\bar{i} + \bar{j} + \bar{k})$.

If the projection of $\bar{a}$ on $\bar{b}+\bar{c}$ is twice the projection of $\bar{b}+\bar{c}$ on $\bar{a}$,and if $|\bar{b}|=2 \sqrt{2}$,$|\bar{c}|=4$,and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{4}$,then $|\bar{a}|=$

If the angle between the two vectors $\vec{u} = \hat{i} + \hat{k}$ and $\vec{v} = \hat{i} - \hat{j} + a\hat{k}$ is $\pi/3$,find the value of $a$.