$A$ unit vector perpendicular to vector $c$ and coplanar with vectors $a$ and $b$ is

Given three unit vectors $a, b, c$ such that $a \perp b$ and $a \parallel c$,then $a \times (b \times c)$ is

Let $\vec{v}$ be a vector such that $\vec{v} \times ((\hat{i}-\hat{k}) \times ((3\hat{i}+4\hat{j}) \times (\hat{j}+\hat{k}))) = \vec{0}$. Suppose $\vec{v} \cdot \hat{j} = -7$. Then $\vec{v} \cdot \hat{i}$ is

Let $\vec{a}=-5 \hat{i}+\hat{j}-3 \hat{k}$,$\vec{b}=\hat{i}+2 \hat{j}-4 \hat{k}$ and $\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$. Then $\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})$ is equal to

Let $\vec{a} = \hat{i} + \hat{j}$, $\vec{c} = \hat{i} - \hat{j}$ and a vector $\vec{b}$ be such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 3$. Then $|\vec{b}| = $

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between vectors $\overline{b}$ and $\overline{c}$,then the value of $\sin \theta$ is