If $$|\bar{a}|=2,|\bar{b}|=3,|\bar{c}|=5$$ and each of the angles between the vectors $$\bar{a}$$ and $$\bar{b}, \bar{b}$$ and $$\bar{c}$$, $$\bar{c}$$ and $$\bar{a}$$ is $$60^{\circ}$$, then the value of $$|\bar{a}+\bar{b}+\bar{c}|$$ is

Let $$\overline{\mathrm{u}}, \overline{\mathrm{v}}$$ and $$\overline{\mathrm{w}}$$ be the vectors such that $$|\overline{\mathrm{u}}|=1; |\bar{v}|=2 ;|\bar{w}|=3$$. If the projection of $$\bar{v}$$ along $$\bar{u}$$ is equal to that of $$\overline{\mathrm{w}}$$ along $$\overline{\mathrm{u}}$$ and $$\overline{\mathrm{v}}, \overline{\mathrm{w}}$$ are perpendicular to each other, then $$|\bar{u}-\bar{v}+\bar{w}|$$ is equal to

Let $$\bar{a}=\hat{i}+2 \hat{j}-\hat{k}$$ and $$\bar{b}=\hat{i}+\hat{j}-\hat{k}$$ be two vectors. If $$\bar{c}$$ is a vector such that $$\bar{b} \times \bar{c}=\bar{b} \times \bar{a}$$ and $$\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}=0$$, then $$\overline{\mathrm{c}} \cdot \overline{\mathrm{b}}$$ is

If $$|\vec{a}|=\sqrt{3} ;|\vec{b}|=5 ; \bar{b} \cdot \bar{c}=10$$, angle between $$\overline{\mathrm{b}}$$ and $$\overline{\mathrm{c}}$$ is $$\frac{\pi}{3}, \overline{\mathrm{a}}$$ is perpendicular to $$\overline{\mathrm{b}} \times \overline{\mathrm{c}}$$. Then the value of $$|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|$$ is