The value of $x$ for which the volume of the parallelepiped formed by the vectors $\vec{a} = \hat{i} + x\hat{j} + \hat{k}$, $\vec{b} = \hat{j} + x\hat{k}$, and $\vec{c} = x\hat{i} + \hat{k}$ is minimum, is

For non-zero vectors $\vec{a}, \vec{b}, \vec{c}$,the condition $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$ holds if and only if:

Unit vectors $a, b, c$ are coplanar. $A$ unit vector $d$ is perpendicular to the given vectors. If $(a \times b) \times (c \times d) = \frac{1}{6}i - \frac{1}{3}j + \frac{1}{3}k$ and the angle between $a$ and $b$ is $30^{\circ}$,then $c = ....$

The value of $(\vec{a} + 2\vec{b} - \vec{c}) \cdot \{(\vec{a} - \vec{b}) \times (\vec{a} - \vec{b} - \vec{c})\}$ is equal to

If $a \cdot b = b \cdot c = c \cdot a = 0$,then what is the value of the scalar triple product $[a b c]$?

$|(a \times b) \cdot c| = |a| |b| |c|$,if