If a vector $\vec{v}=3 \hat{i}$ is rotated in the $x-z$ plane by an angle $\theta$ with respect to $x$－axis in the clockwise direction，then for an observer at $+y$ axis the vector will be

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $|\vec{a}|=1,|\vec{b}|=2$ and $|\vec{c}|=4$ along with $\vec{a}+\vec{b}+\vec{c}=0$ ．Then，the value of $4 \vec{a} \cdot \vec{b}+3 \vec{b} \cdot \vec{c}+3 \vec{c} \cdot \vec{a}$ will be

Let $$\theta$$ be the angle between two vectors $$\vec{A}$$ and $$\vec{B}$$. If $$\hat{a}_{\perp}$$ is the unit vector perpendicular to $$\vec{A}$$, then the direction of $$ \overrightarrow{\mathrm{B}}-\mathrm{B} \sin \theta \hat{\mathrm{a}}_{\perp} \text { is }$$