If $O$ is the circumcentre and $O'$ is the orthocentre of the triangle $ABC$,then $\overrightarrow{O'A} + \overrightarrow{O'B} + \overrightarrow{O'C} = $

If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$,then find $\theta$ and the components of $\vec{a}$.

If $L, M, N$ are the midpoints of the sides $PQ, QR$ and $RP$ of $\triangle PQR$ respectively,then $\overrightarrow{QM} + \overrightarrow{LN} + \overrightarrow{ML} + \overrightarrow{RN} - \overrightarrow{MN} - \overrightarrow{QL} = $

If the position vectors of $A$ and $B$ are respectively $(1, 1, 0)$ and $(0, 1, 1)$,then $\overrightarrow{AB} =$

The components of a vector $\vec{a}$ with respect to a rectangular Cartesian system are $2p$ and $1$. The system is rotated about the origin through a certain angle in the counter-clockwise direction. If the components of $\vec{a}$ with respect to the new system are $p + 1$ and $1$,then:

Find the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$.