The position vector of the point which divides the line segment joining the points with position vectors $2a - 3b$ and $3a - 2b$ internally in the ratio $2 : 3$ is:

$A$ vector coplanar with the non-collinear vectors $\vec{a}$ and $\vec{b}$ is:

If $P$ divides the line segment joining the points $A$ and $B$ in the ratio $2:1$ and the position vectors of $A$ and $B$ are $\hat{i}-2\hat{j}$ and $-3\hat{i}+5\hat{j}$ respectively,then the position vector of $P$ is

$\hat{i}-2 \hat{j}+\hat{k}$, $2 \hat{i}+\hat{j}-\hat{k}$, and $\hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of the vertices $A, B$, and $C$ of a triangle $ABC$ respectively. If $D$ and $E$ are the midpoints of $BC$ and $CA$ respectively, then the unit vector along $\overrightarrow{DE}$ is

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

Find a vector in the direction of vector $\vec{a} = \hat{i} - 2\hat{j}$ that has magnitude $7$ units.