The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda \hat{j} + 2\hat{k}$, and $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda \hat{k}$ are coplanar is:

Let $\vec{a}$ be a vector which is perpendicular to the vector $3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k}$. If $\vec{a} \times (2 \hat{i} + \hat{k}) = 2 \hat{i} - 13 \hat{j} - 4 \hat{k}$,then the projection of the vector $\vec{a}$ on the vector $2 \hat{i} + 2 \hat{j} + \hat{k}$ is

If $|\vec{a}| = 4, |\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 8$, then the scalar triple product $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} \times \vec{b}]$ is equal to:

Let $a=2 \hat{i}+\hat{j}-3 \hat{k}$ and $b=\hat{i}+3 \hat{j}+2 \hat{k}$. Then the volume of the parallelopiped having coterminous edges as $a, b$ and $c$, where $c$ is the vector perpendicular to the plane of $a, b$ and $|c|=2$ is

If the volume of the parallelepiped is $158 \text{ cubic units}$,whose coterminous edges are given by the vectors $\bar{a} = (\hat{i} + \hat{j} + n \hat{k})$,$\bar{b} = (2 \hat{i} + 4 \hat{j} - n \hat{k})$,and $\bar{c} = (\hat{i} + n \hat{j} + 3 \hat{k})$,where $n \geq 0$,then the value of $n$ is:

Let $\vec{OD} = \hat{i} + 2\hat{j} + 6\hat{k}$ and $\vec{CB} = -3\hat{i} - 2\hat{k}$ be the diagonals of the parallelogram $OBDC$. If $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$, then the volume of the parallelepiped determined by vectors $\vec{OA}, \vec{OB}$, and $\vec{OC}$ (in cubic units) is: