If three unit vectors $a, b, c$ are such that $a \times (b \times c) = \frac{b}{2},$ then the vector $a$ makes with $b$ and $c$ respectively the angles

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

The vectors $\bar{a}$ and $\bar{b}$ are not perpendicular and $\overline{c}$ and $\overline{d}$ are two vectors satisfying $\overline{b} \times \overline{c} = \overline{b} \times \overline{d}$ and $\overline{a} \cdot \overline{d} = 0$. Then the vector $\overline{d}$ 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:

Let $\vec{v}$ be a unit vector which follows the equation $\vec{v} \times \vec{b} = \vec{c}$. Also,$|\vec{b}| = 2$ and $|\vec{c}| = \sqrt{3}$. Then,which of the following is true?

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors,then $\frac{\vec{a} \times(\vec{b} \times \vec{a})}{|\vec{a}|^2}$ represents: