$A$ force of magnitude $6$ acts along the vector $(9, 6, -2)$ and passes through a point $A(4, -1, -7)$. The moment of the force about the point $O(1, -3, 2)$ 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 $9k^2 =$

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:

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

In the above figure,$P$ divides $AC$ in the ratio $3:4$ and $Q$ divides $BC$ in the ratio $4:3$. Then $M$ divides $AQ$ in the ratio:

If $\vec{l}, \vec{m}, \vec{n}$ are coplanar vectors,for what value of $\lambda$ are the points with position vectors $\vec{l} - 2\vec{m} + 3\vec{n}$,$2\vec{l} + \lambda\vec{m} - 4\vec{n}$,and $-7\vec{m} + 10\vec{n}$ collinear?