If $a = i + 2j - 2k$,$b = 2i - j + k$ and $c = i + 3j - k$,then $a \times (b \times c)$ is equal to

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be given as $\vec{a} = a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$,$\vec{b} = b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$,and $\vec{c} = c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}$. Then show that $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$.

$A$ non-zero vector $\vec{a}$ is parallel to the line of intersection of the plane defined by $\hat{i}$ and $\hat{i} + \hat{j}$,and the plane defined by $\hat{i} - \hat{j}$ and $\hat{i} + \hat{k}$. Find the angle between $\vec{a}$ and $\hat{i} - 2\hat{j} + 2\hat{k}$.

Let $a=2 \hat{i}-2 \hat{j}+\hat{k}$ and $b=-\hat{j}+\hat{k}$. 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 $\frac{\pi}{3}$,then $|(a \times b) \times c|=$

Direction ratios of the line which is perpendicular to the lines with direction ratios $-1, 2, 2$ and $0, 2, 1$ are

The direction ratios of the line perpendicular to the lines having direction ratios $2, 3, 1$ and $1, 2, 1$ are