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

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors such that $$|\bar{a}+\bar{b}+\bar{c}|=1, \overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$$ and $$|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mathrm{b}}|=\frac{1}{\sqrt{2}},|\overline{\mathrm{c}}|=\frac{1}{\sqrt{6}}$$, then the angle between $$\bar{a}$$ and $$\bar{b}$$ is

Let $$\bar{a}, \bar{b}, \bar{c}$$ be three vectors such that $$|\bar{a}|=\sqrt{3}, |\bar{b}|=5, \bar{b} \cdot \bar{c}=10$$ and the angle between $$\bar{b}$$ and $$\bar{c}$$ is $$\frac{\pi}{3}$$. If $$\bar{a}$$ is perpendicular to the vector $$\bar{b} \times \bar{c}$$, then $$|\bar{a} \times(\bar{b} \times \bar{c})|$$ is equal to