If $a, b, c$ are non-coplanar vectors and $d = \lambda a + \mu b + \nu c$,then $\lambda = \dots$

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar vectors and $\overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}$ are defined by the relations $\overrightarrow{p}=\frac{\overrightarrow{b} \times \overrightarrow{c}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}, \quad \overrightarrow{q}=\frac{\overrightarrow{c} \times \overrightarrow{a}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a} \times \overrightarrow{b}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}$,then $\overrightarrow{a} \cdot \overrightarrow{p}+\overrightarrow{b} \cdot \overrightarrow{q}+\overrightarrow{c} \cdot \overrightarrow{r}$ is equal to

Let $\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$,$\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}$,and $\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\vec{a} \times \vec{c}$ is:

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

The value of $a$ such that the volume of the parallelepiped formed by the vectors $i + aj + k$,$j + ak$,and $ai + k$ is minimum is:

Find the volume of a tetrahedron whose vertices are given by the vectors $-i + j + k$,$i - j + k$,and $i + j - k$,with the fourth vertex being the origin.