If $\vec{x} \cdot \vec{y} = 0$ then,$(\vec{y} \times \vec{x}) \times \vec{x} = $ . . . . . . . where,$|\vec{x}| = 1$.

Let $\vec{a} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. If $\vec{a} \times \vec{b} + \vec{c} = \vec{0}$ and $\vec{a} \cdot \vec{b} = 3$,find the vector $\vec{b}$.

If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} \times (\vec{a} \times \hat{i}) + \hat{j} \times (\vec{a} \times \hat{j}) + \hat{k} \times (\vec{a} \times \hat{k})$,then $|\vec{b}|$ is

$a \times (b \times c) + b \times (c \times a) + c \times (a \times b) =$

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors of magnitudes $\sqrt{3}, 1, 2$ respectively,such that $\vec{a} \times (\vec{a} \times \vec{c}) + 3\vec{b} = \vec{0}$. If $\theta$ is the angle between $\vec{a}$ and $\vec{c}$,then $\cos^2 \theta = $

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(\overline{a}-2 \overline{b}) \cdot \{(\overline{a} \times \overline{b}) \times (2 \overline{a}+\overline{b})\}$ is