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

The vectors $a$ and $b$ are non-collinear. The value of $x$ for which the vectors $c = (x - 2)a + b$ and $d = (2x + 1)a - b$ are collinear,is

$A$ girl walks $4 \, km$ towards west,then she walks $3 \, km$ in a direction $30^{\circ}$ east of north and stops. Determine the girl's displacement from her initial point of departure.

If $\overline{a}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}$,$\overline{b}=4 \hat{\imath}+5 \hat{\jmath}+3 \hat{k}$ and $\overline{c}=6 \hat{\imath}+\hat{\jmath}+5 \hat{k}$ are the position vectors of the vertices of a triangle $ABC$ respectively,then the position vector of the intersection of the medians (centroid) of the triangle $ABC$ is:

Two vectors $u$ and $v$ are parallel if and only if

The diagonals of a parallelogram are the vectors $\vec{d_1} = 3 \hat{i} + 6 \hat{j} - 2 \hat{k}$ and $\vec{d_2} = -\hat{i} - 2 \hat{j} - 8 \hat{k}$. Then the length of the shorter side of the parallelogram is