If the position vectors of the vertices of a triangle are $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} - 3\hat{j} - 5\hat{k}$,and $3\hat{i} - 4\hat{j} - 4\hat{k}$,then the triangle is:

Let $\bar{a} = \bar{i} + 2\bar{j} + 2\bar{k}$ and $\bar{b} = 2\bar{i} - \bar{j} + p\bar{k}$ be two vectors. If the angle between $\bar{a}$ and $\bar{b}$ is $60^{\circ}$, then $p =$

Let $ABC$ be a triangle. Let a point $P$ divide $AB$ in the ratio $1:2$ internally and a point $Q$ divide $BC$ in the ratio $1:2$ internally. Let $D$ be the point of intersection of $AQ$ and $CP$. If the area of the triangle $ABC$ is $k$ square units,then the area of the triangle $BCD$ in square units is:

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be vectors of magnitude $2, 3$ and $4$ respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c})$,$\bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\bar{c}$ is perpendicular to $(\bar{a}+\bar{b})$,then the magnitude of $\bar{a}+\bar{b}+\bar{c}$ is

If $\vec{OA} = 2\hat{i} + 2\hat{j} + \hat{k}$,$\vec{OB} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ and the length of the internal bisector of $\angle BOA$ of triangle $AOB$ is $k$,then $9k^2 =$

For vectors $\vec{a}, \vec{b}$ and $\vec{c}$,if $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=5$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . .