If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + y\hat{j} + z\hat{k}$,find the number of possible vectors $\vec{b}$ such that $\vec{a} \cdot \vec{b} = 10$,where $(x, y, z) \in \mathbb{N}$.

$3 \hat{i}-2 \hat{j}-\hat{k}, -2 \hat{i}-\hat{j}+3 \hat{k}$ and $-\hat{i}+3 \hat{j}-2 \hat{k}$ are the position vectors of the vertices $A, B$ and $C$ of a $\triangle ABC$ respectively. If $H$ is its orthocenter,then $\overrightarrow{HA}+\overrightarrow{HB}+\overrightarrow{HC} = $

The least positive integral value of $\alpha$,for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute,is

The projection vector of $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ on $\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}$ is . . . . . . .

If $a$ and $b$ are two unit vectors such that $a+2b$ and $5a - 4b$ are perpendicular to each other,then the angle between $a$ and $b$ is ............. $^o$

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