If $\bar{a}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\bar{b}=2 \hat{i}+3 \hat{j}-\hat{k}$,then the angle between the vectors $(2 \bar{a}+\bar{b})$ and $(\bar{a}+2 \bar{b})$ is

If $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k},$ then show that the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are perpendicular.

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2$. If $\theta \in(0, \pi)$ is the angle between $\hat{a}$ and $\hat{b}$,then among the statements:
$(S_{1})$: $2|\hat{a} \times \hat{b}|=|\hat{a}-\hat{b}|$
$(S_{2})$: The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

If vector $a$ has magnitude $5$ and points north-east,and vector $b$ has magnitude $5$ and points north-west,then $|a - b| = $

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that $\overline{OP} \cdot \overline{OQ} + \overline{OR} \cdot \overline{OS} = \overline{OR} \cdot \overline{OP} + \overline{OQ} \cdot \overline{OS} = \overline{OQ} \cdot \overline{OR} + \overline{OP} \cdot \overline{OS}$. Then the triangle $PQR$ has $S$ as its

If $A(3,2,3)$,$B(1,4,6)$,and $C(7,4,5)$ are the three vertices of a parallelogram $ABCD$,then the angle between its diagonal through $D$ and the side $DC$ is