If $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$,$\vec{b} = \hat{i} + \hat{j} - 2\hat{k}$,and $\vec{c} = \hat{i} + 3\hat{j} - (\lambda^2 + 3\lambda)\hat{k}$ (where $\lambda$ is a constant) and $\vec{a}$ is perpendicular to $\vec{c} - \lambda\vec{b}$,then the sum of different values of $\lambda$ is:

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. Let $\overline{c}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\bar{a} \times \bar{b}) \times \bar{c}|=3$ and the angle between $\overline{c}$ and $\overline{a} \times \overline{b}$ is $30^{\circ}$,then $\overline{a} \cdot \overline{c}$ is equal to

If $c = 2 \lambda (a \times b) + 3 \mu (b \times a)$ where $a \times b \neq 0$ and $c \cdot (a \times b) = 0$,then:

If $\vec{a} = 2\hat{i} - \hat{j} - 2\hat{k}$ and $\vec{b} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ are two vectors,and we consider a vector $\vec{c} = \vec{a} + t\vec{b}$,find the value of $t$ such that the magnitude $|\vec{c}|$ is minimum.

If $|a \cdot b| = 3$ and $|a \times b| = 4$,then the angle between $a$ and $b$ is

If the points $P$ and $Q$ are respectively the circumcentre and the orthocentre of a $\triangle ABC$,then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ is equal to