If $|\vec{a}| = |\vec{b}| = 1$, $|\vec{c}| = 2$ and $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is ...

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero and non-coplanar vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b}}{2}$, then the angle between $\vec{a}$ and $\vec{b}$ is ...

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 $\vec{a}$ and $\vec{b}$ are perpendicular,then $\vec{a} \times(\vec{a} \times(\vec{a} \times(\vec{a} \times \vec{b})))$ is equal to

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

For unit vectors $\bar{a}, \bar{b}, \bar{c}$,if $\bar{a} \times (\bar{b} \times \bar{c}) = \frac{\bar{b}}{2}$ and $\bar{b}, \bar{c}$ are non-collinear vectors,then the angles made by $\bar{a}$ with $\bar{b}$ and $\bar{c}$ respectively are: