In a quadrilateral $ABCD$, the point $P$ divides $DC$ in the ratio $1:3$ internally and $Q$ is the mid-point of $AC$. If $\vec{AB} + \vec{AD} + \vec{BC} - 2\vec{DC} = \lambda \vec{PQ}$, then the value of $\lambda$ is

If vectors $a$ and $b$ represent two consecutive sides of a regular hexagon,then the vectors representing the remaining four sides in order are:

The sum of two forces is $18 \ N$ and the resultant,whose direction is at right angles to the smaller force,is $12 \ N$. The magnitudes of the two forces are:

If a parallelogram is constructed on the vectors $\vec{a} = 3\vec{p} - \vec{q}$ and $\vec{b} = \vec{p} + 3\vec{q}$, where $|\vec{p}| = 3$, $|\vec{q}| = 2$ and the angle between $\vec{p}$ and $\vec{q}$ is $\pi/3$, then the ratio of the lengths of adjacent sides $|\vec{a}|$ and $|\vec{b}|$ of the parallelogram is:

If $\vec{a}$ and $\vec{b}$ are the position vectors of $A$ and $B$,respectively,find the position vector of a point $C$ in $BA$ produced such that $BC = 1.5 BA$.

If the position vectors of points $A$ and $B$ are $2\hat{i} + 3\hat{j} + 4\hat{k}$ and $3\hat{i} - 4\hat{j} - 5\hat{k}$ respectively,find $\overline{AB}$.