If $(\bar{a} \times \bar{b}) \times \bar{c} = -5 \bar{a} + 4 \bar{b}$ and $\bar{a} \cdot \bar{b} = 3$,then the value of $\bar{a} \times (\bar{b} \times \bar{c})$ 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:

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$,$\vec{b}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c}=5$,then $|\vec{c}|$ is equal to

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=-\hat{i}+2\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$,then $(\vec{a}-\vec{b}) \cdot [(\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c})]$ is

For any vector $r$,the expression $i \times(r \times i) + j \times(r \times j) + k \times(r \times k)$ is equal to:

Let $\vec{a}$ be a non-zero vector. If $\vec{x}=\hat{i} \times(\vec{a} \times \hat{i})$,$\vec{y}=\hat{j} \times(\vec{a} \times \hat{j})-\vec{a}$ and $\vec{z}=\hat{k} \times(\vec{a} \times \hat{k})-\vec{a}$,then $\left[\begin{array}{lll}\vec{x} & \vec{y} & \vec{z}\end{array}\right]=$