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:

The volume of the tetrahedron whose vertices are $A(-1, 2, 3)$, $B(3, -2, 1)$, $C(p, 1, 3)$, and $D(-1, -2, 4)$ is $\frac{16}{3}$ cubic units. Find the value of $p$.

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors,then $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} = \dots$

If $\vec{a}, \vec{b}, \vec{c}$ are non-zero and non-coplanar vectors such that $(\vec{a} + \lambda \vec{b}) \cdot [(\vec{b} + 3\vec{c}) \times (\vec{c} - 4\vec{a})] = 0$,then $\lambda$ is equal to

If $[\vec{p}-\vec{r}, \vec{q}, \vec{s}] + [\vec{p}+\vec{q}, \vec{r}, \vec{s}] = m[\vec{p}, \vec{r}, \vec{s}] + n[\vec{q}, \vec{r}, \vec{s}] + t[\vec{p}, \vec{q}, \vec{s}]$,then the values of $m$,$n$,$t$ respectively are . . . . . .

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$,$\vec{b}=\hat{i}+3 \hat{j}-\hat{k}$ and $\vec{c}=3 \hat{i}-\hat{j}-2 \hat{k}$,then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c}\end{array}\right|$ is: