If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + aj + 5k$ are coplanar,find the value of $a$.

If the volume of a tetrahedron,whose vertices are with position vectors $\hat{i}-6 \hat{j}+10 \hat{k}$,$-\hat{i}-3 \hat{j}+7 \hat{k}$,$5 \hat{i}-\hat{j}+\lambda \hat{k}$ and $7 \hat{i}-4 \hat{j}+7 \hat{k}$ is $11$ cubic units,then the value of $\lambda$ is:

The volume of the parallelepiped determined by the vectors $\vec{a} + \vec{b}, \vec{b} + \vec{c}$ and $\vec{c} + \vec{a}$ is $4$. Then the volume of the parallelepiped determined by the vectors $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ is:

The altitude of the parallelepiped,whose coterminus edges are the vectors $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}-\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}+3\hat{k}$,where $\bar{a}$ and $\bar{b}$ are the sides of the base of the parallelepiped,is:

The volume of a tetrahedron whose vertices are $A \equiv (-1, 2, 3)$,$B \equiv (3, -2, 1)$,$C \equiv (2, 1, 3)$,and $D \equiv (-1, -2, 4)$ is

If $\vec{a}=\alpha \hat{i}+\beta \hat{j}+3 \hat{k}$,$\vec{b}=\hat{j}+2 \hat{k}$,and $\vec{c}=3 \hat{i}+2 \hat{j}+\hat{k}$ are linearly dependent vectors and the magnitude of $\vec{a}$ is $\sqrt{14}$. If $\alpha$ and $\beta$ are integers,then $\alpha+\beta=$