If $|a|=3, |b|=4$ and the angle between $a$ and $b$ is $120^{\circ}$,then $|4a+3b|$ is equal to

In a triangle $ABC$ with usual notations,if $|\overline{BC}|=8, |\overline{CA}|=7, |\overline{AB}|=10$,then the projection of $\overline{AB}$ on $\overline{AC}$ is

If $a$ and $b$ are unit vectors and $\theta$ is the angle between $a$ and $b$,then $\sin \frac{\theta}{2}$ is equal to

If $a, b, c$ are three non-zero,non-coplanar vectors and $b_1 = b - \frac{b \cdot a}{|a|^2} a$,$b_2 = b + \frac{b \cdot a}{|a|^2} a$,$c_2 = c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_1}{|b_1|^2} b_1$,$c_3 = c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_2}{|b_2|^2} b_2$,and $c_4 = a - \frac{c \cdot a}{|a|^2} a$. Then which of the following is a set of mutually orthogonal vectors?

If $\bar{a}$ and $\bar{b}$ are two vectors such that $|\bar{a}|=5$,$|\bar{b}|=12$ and $|\bar{a}-\bar{b}|=13$,then $|2\bar{a}+\bar{b}|=$

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{j}-3 \hat{k}$. If $\vec{b}=\vec{c}-\vec{d}$,$\vec{a}$ is parallel to $\vec{c}$,and $\vec{a}$ is perpendicular to $\vec{d}$,then $\vec{c}+\vec{d}=$