If the position vectors of $A, B, C,$ and $D$ are $2i + j,$ $i - 3j,$ $3i + 2j,$ and $i + \lambda j$ respectively and $\overrightarrow{AB} \parallel \overrightarrow{CD},$ then the value of $\lambda$ is:

$L$ and $M$ are two points with position vectors $2 \vec{a}-\vec{b}$ and $\vec{a}+2 \vec{b}$ respectively. The position vector of the point $N$ which divides the line segment $LM$ in the ratio $2:1$ externally is

$OABCD$ is a pentagon in which the sides $OA$ and $CB$ are parallel and the sides $OD$ and $AB$ are parallel. Also,it is given that $\frac{OA}{CB}=2$,$\frac{OD}{AB}=\frac{1}{3}$. If $\vec{OA}=\vec{a}, \vec{OD}=\vec{d}$,then $\vec{AD}+\vec{OC}+\vec{DC}=$

$a, b, c$ are non-coplanar vectors. If $a+3 b+4 c=x(a-2 b+3 c)+y(a+5 b-2 c)+z(6 a+14 b+4 c)$,then $x+y+z=$

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes $OX, OY,$ and $OZ$.

If $a$ and $b$ are two vectors such that $a \cdot b = 0$ and $a \times b = 0$,then .....