The vector$(s)$ which is/are coplanar with vectors $\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k}$,and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}$ is/are:
$(A) \hat{j}-\hat{k}$
$(B) -\hat{i}+\hat{j}$
$(C) \hat{i}-\hat{j}$
$(D) -\hat{j}+\hat{k}$

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The midpoint of the arc $PQ$ is $R$. If $\vec{OP}=\vec{u}$,$\vec{OR}=\vec{v}$ and $\vec{OQ}=\alpha \vec{u}+\beta \vec{v}$,then $\alpha, \beta^2$ are the roots of the equation

If three unit vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$,where $a, b \in \mathbb{Z}$. Let $S$ be the point on $QR$,which is equidistant from the lines $PQ$ and $PR$. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$,then the value of $3a-4b$ is . . . . . . .

If $\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}$ are such that $\vec{a}+\lambda \vec{b}$ is perpendicular to $\vec{c},$ then find the value of $\lambda$.

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$,and $|\vec{a} + \vec{b} + \vec{c}| = 1$,then the angle between $\vec{b}$ and $\vec{c}$ is: