The points $D, E, F$ divide $BC, CA$ and $AB$ of the triangle $ABC$ in the ratio $1:4, 3:2$ and $3:7$ respectively and the point $K$ divides $AB$ in the ratio $1:3$,then $(\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF}) : \overrightarrow{CK}$ is equal to

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors in the $xyz$-space such that $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq 0$. If $A, B, C$ are points with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively,then the number of possible positions of the centroid of $\triangle ABC$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. If $\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ and $\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$,then the ordered pair $(\lambda, \vec{d})$ is equal to:

The vector that is parallel to the vector $2 \hat{i} - 2 \hat{j} - 4 \hat{k}$ and coplanar with the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is

If $\vec{a}$ and $\vec{b}$ represent the two adjacent sides $\vec{AB}$ and $\vec{BC}$ of a regular hexagon $ABCDEF$,then $\vec{AE} = \dots$

Let $\triangle PQR$ be a triangle. Let $\vec{a}=\overline{QR}, \vec{b}=\overline{RP}$ and $\vec{c}=\overline{PQ}$. If $|\vec{a}|=12, |\vec{b}|=4\sqrt{3}$ and $\vec{b} \cdot \vec{c}=24$,then which of the following is (are) true?
$(A) \frac{|\vec{c}|^2}{2}-|\vec{a}|=12$
$(B) \frac{|\vec{c}|^2}{2}+|\vec{a}|=30$
$(C) |\vec{a} \times \vec{b}+\vec{c} \times \vec{a}|=48\sqrt{3}$
$(D) \vec{a} \cdot \vec{b}=-72$