If $\overrightarrow x = 3\hat i - 6\hat j - \hat k$,$\overrightarrow y = \hat i + 4\hat j - 3\hat k$ and $\overrightarrow z = 3\hat i - 4\hat j - 12\hat k$,then the magnitude of the projection of $\overrightarrow x \times \overrightarrow y$ on $\overrightarrow z$ is

If the volume of the parallelepiped with $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ as coterminous edges is $40 \text{ cubic units}$,then the volume of the parallelepiped having $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ as coterminous edges in cubic units is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}$,$x \in R$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a} \cdot \vec{d}=1$,then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

If $a, b, c$ are non-negative distinct numbers and $a \hat{\imath}+a \hat{\jmath}+c \hat{k}$,$\hat{\imath}+\hat{k}$ and $c \hat{\imath}+c \hat{\jmath}+b \hat{k}$ are coplanar vectors,then

If $a$ is perpendicular to $b$ and $c$,$|a| = 2$,$|b| = 3$,$|c| = 4$ and the angle between $b$ and $c$ is $\frac{2\pi}{3}$,then $[a \; b \; c]$ is equal to (in $\sqrt{3}$):

The maximum value and minimum value of the volume of the parallelepiped having coterminous edges $\hat{i}+x \hat{j}+\hat{k}$,$\hat{j}+x \hat{k}$,and $x \hat{i}+\hat{k}$ are respectively: