Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{c} = \hat{j} - \hat{k}$ and a vector $\vec{b}$ be such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 3$. Then $|\vec{b}|$ equals?

$\vec{c}$ is a vector along the bisector of the internal angle between the vectors $\vec{a}=4 \hat{i}+7 \hat{j}-4 \hat{k}$ and $\vec{b}=12 \hat{i}-3 \hat{j}+4 \hat{k}$. If the magnitude of $\vec{c}$ is $3 \sqrt{13}$,then $\vec{c}=$

Let the position vectors of points $A$ and $B$ be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k},$ respectively. $A$ point $P$ divides the line segment $AB$ internally in the ratio $\lambda:1$ $(\lambda>0)$. If $O$ is the origin and $\overrightarrow{OB} \cdot \overrightarrow{OP}-3|\overrightarrow{OA} \times \overrightarrow{OP}|^{2}=6,$ then $\lambda$ is equal to

If $\triangle ABC$ is right-angled at $A$,where $A \equiv (4, 2, x)$,$B \equiv (3, 1, 8)$,and $C \equiv (2, -1, 2)$,then the value of $x$ is

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

In a triangle $ABC,$ if $|\overline{BC}|=8, |\overline{CA}|=7, |\overline{AB}|=10,$ then the projection of the vector $\overline{AB}$ on $\overline{AC}$ is equal to ....... .