If $a \times (b \times c) = 0,$ then

The number of vectors of unit length perpendicular to the two vectors $a=(1,1,0)$ and $b=(0,1,1)$ is

Let $a = 2i + j - 2k$ and $b = i + j$. If $c$ is a vector such that $a \cdot c = |c|$,$|c - a| = 2\sqrt{2}$,and the angle between $(a \times b)$ and $c$ is $30^{\circ}$,then $|(a \times b) \times c| = \dots$

Two adjacent sides of a parallelogram are given by vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$. Find the area of the parallelogram in square units.

If $a = 2i - 3j - k$ and $b = i + 4j - 2k$,then $a \times b$ is

Find a vector that is perpendicular to both vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + \hat{j}$.