Let $$\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}$$ and $$\bar{b}=\hat{i}+\hat{j}$$. If $$\bar{c}$$ is a vector such that $$\bar{a} \cdot \bar{c}=|\bar{c}|,|\bar{c}-\bar{a}|=2 \sqrt{2}$$ and the angle between $$\bar{a} \times \bar{b}$$ and $$\bar{c}$$ is $$\frac{2 \pi}{3}$$, then $$|(\bar{a} \times \bar{b}) \times \bar{c}|=$$

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