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

If $\overline{a}, \overline{b}, \overline{c}$ are mutually perpendicular vectors having magnitudes $1, 2, 3$ respectively,then the value of $[\overline{a}+\overline{b}+\overline{c} \quad \overline{b}-\overline{a} \quad \overline{c}]$ is

The maximum volume of a parallelepiped (in cubic units) with vectors $\vec{u} = 2a\hat{i} + \hat{k}$, $\vec{v} = a\hat{j} - a\hat{k}$, and $\vec{w} = 3\hat{i} + a\hat{j}$, where $a \in [0, 1]$, as its coterminous edges is:

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

If $p = \frac{b \times c}{[a, b, c]}, q = \frac{c \times a}{[a, b, c]}, r = \frac{a \times b}{[a, b, c]}$,where $a, b, c$ are three non-coplanar vectors,then the value of $(a + b + c) \cdot (p + q + r)$ is given by

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