If $\vec{a}$ and $\vec{b}$ are vectors in space given by $\vec{a}=\frac{\hat{i}-2 \hat{j}}{\sqrt{5}}$ and $\vec{b}=\frac{2 \hat{i}+\hat{j}+3 \hat{k}}{\sqrt{14}}$,then the value of $(2 \vec{a}+\vec{b}) \cdot[(\vec{a} \times \vec{b}) \times(\vec{a}-2 \vec{b})]$ is

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 = $

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is

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:

If $\vec{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + 0\hat{j} + \hat{k}$,find a vector $\vec{c}$ satisfying the following conditions:
$(i)$ $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$.
$(ii)$ $\vec{c}$ is perpendicular to $\vec{b}$.
$(iii)$ $\vec{a} \cdot \vec{c} = 7$.

Statement $(A)$ : If $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$,then $\vec{a} \times (\vec{b} \times \vec{c}) = 0$.
Reason $(R)$ : If $\vec{b}$ is perpendicular to $\vec{c}$,then $\vec{b} \times \vec{c} = 0$.