$a, b, c$ are three vectors such that $a + b + c = 0$,$|a| = 1, |b| = 2, |c| = 3$. Then $a \cdot b + b \cdot c + c \cdot a$ is equal to:

In triangle $ABC$,the point $P$ divides $BC$ internally in the ratio $3:4$ and $Q$ divides $CA$ internally in the ratio $5:3$. If $AP$ and $BQ$ intersect in a point $G$,then $G$ divides $AP$ internally in the ratio

Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if $\theta =$

If the position vectors of $A, B, C, D$ are $\bar{i}+2\bar{j}+2\bar{k}, 2\bar{i}-\bar{j}, \bar{i}+\bar{j}+3\bar{k}$ and $4\bar{j}+5\bar{k}$ respectively,then the quadrilateral $ABCD$ is a

If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is

If $a$ and $b$ are two unit vectors and $\theta$ is the angle between them,then the unit vector along the angular bisector of $a$ and $b$ is given by