Let $a, b, c$ be three vectors. Then $a \times (b \times c) = (a \times b) \times c$ if:

Let $\vec{a}$ be a unit vector and $\vec{b}$ be a nonzero vector not parallel to $\vec{a}$. The angles of the triangle,two of whose sides are represented by $\sqrt{3}(\vec{a} \times \vec{b})$ and $\vec{b} - (\vec{a} \cdot \vec{b})\vec{a}$,are

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha \vec{b} - \hat{n}, (\alpha \neq 0)$ and $\vec{b} \cdot \vec{c} = 12$,then $|\vec{c} \times (\vec{a} \times \vec{b})|$ is equal to:

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and $\vec{a} \cdot \vec{d}=18$,then $|\vec{a} \times \vec{d}|^2$ is equal to $..........$.

If $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$,where $\vec{a}, \vec{b},$ and $\vec{c}$ are any three vectors such that $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$,then $\vec{a}$ and $\vec{c}$ are:

The position vector of a point $P$ is $2 \hat{i}+\hat{j}+3 \hat{k}$ and $a=-\hat{i}-2 \hat{k}, b=\hat{i}+\hat{j}+2 \hat{k}$ are two vectors which determine a plane $\pi$. The equation of a line through $P$ normal to $b$ and lying on the plane $\pi$ is