If $a = i + j + k$,$b = i + j$,$c = i$ and $(a \times b) \times c = \lambda a + \mu b$,then $\lambda + \mu = \dots$

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 $\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$,$\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$,and $\vec{c} = \hat{i} + 3\hat{j} - \hat{k}$,then find $\vec{a} \times (\vec{b} \times \vec{c})$.

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three unit vectors such that $\overline{a} \times(\overline{b} \times \overline{c})=\frac{\sqrt{3}}{2}(\overline{b}+\overline{c})$. If $\overline{b}$ is not parallel to $\overline{c}$,then the angle between $\overline{a}$ and $\overline{b}$ is

The position vector of a point $P$ is $2 \hat{i}+\hat{j}+3 \hat{k}$ and $a=-\hat{i}-2 \hat{k}, b=\hat{i}+\hat{j}+2 \hat{k}$ are two vectors which determine a plane $\pi$. The equation of a line through $P$ normal to $b$ and lying on the plane $\pi$ is

The unit vector which is orthogonal to the vector $5 \hat{i}+2 \hat{j}+6 \hat{k}$ and is coplanar with the vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is