If $D, E$ and $F$ are the mid-points of the sides $BC, CA$ and $AB$ of triangle $ABC$ respectively,then $\overline{AD} + \frac{2}{3} \overline{BE} + \frac{1}{3} \overline{CF} =$

The vector having magnitude of $2 \sqrt{29}$ units in the direction of vector $\vec{a} = 4 \hat{i} + 3 \hat{j} - 2 \hat{k}$ is . . . . . . .

If $m_1, m_2, m_3$ and $m_4$ are respectively the magnitudes of the vectors $\overrightarrow{a}_1=2 \hat{i}-\hat{j}+\hat{k}$,$\overrightarrow{a}_2=3 \hat{i}-4 \hat{j}-4 \hat{k}$,$\overrightarrow{a}_3=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{a}_4=-\hat{i}+3 \hat{j}+\hat{k}$,then the correct order of $m_1, m_2, m_3$ and $m_4$ is

If $C$ is the midpoint of $AB$ and $P$ is any point outside $AB$,then

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:

If the position vector of one end of the line segment $AB$ is $2\hat{i} + 3\hat{j} - \hat{k}$ and the position vector of its midpoint is $3\,(\hat{i} + \hat{j} + \hat{k}),$ then the position vector of the other end is