If $\bar{a}=\hat{i}+2 \hat{j}-3 \hat{k}$,$\bar{b}=3 \hat{i}-\hat{j}+2 \hat{k}$,$\bar{c}=\hat{i}+3 \hat{j}+\hat{k}$ and $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$,then $\lambda=$

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

If $a, b, c$ are distinct real numbers and $P, Q, R$ are three points whose position vectors are respectively $a \hat{i}+b \hat{j}+c \hat{k}$,$b \hat{i}+c \hat{j}+a \hat{k}$ and $c \hat{i}+a \hat{j}+b \hat{k}$,then $\angle Q P R=$

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}|=7$,$|\vec{b}|=1$ and $|\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2$,then the values of $k$ and $\theta$ are

If $\theta$ is an obtuse angle between vectors $\overline{a}$ and $\overline{b}$ such that $|\overline{a}|=5$,$|\overline{b}|=3$ and $|\overline{a} \times \overline{b}|=5 \sqrt{5}$,then $\overline{a} \cdot \overline{b}=$

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$,then $|\vec{a} \times \vec{c}|$ is equal to