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

If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are unit vectors and $$\theta$$ is angle between $$\overline{\mathrm{a}}$$ and $$\bar{c}$$ and $$\bar{a}+2 \bar{b}+2 \bar{c}=\overline{0}$$, then $$|\bar{a} \times \bar{c}|=$$

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors with magnitudes $$\sqrt{3}$$, 1, 2 respectively, such that $$\bar{a} \times(\bar{a} \times \bar{c})+3 \bar{b}=\overline{0}$$, if $$\theta$$ is the angle between $$\bar{a}$$ and $$\bar{c}$$, then $$\sec ^2 \theta$$ is

If the vectors $$p \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+q \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+r \hat{\mathbf{k}}(p \neq q \neq r \neq 1)$$ are coplanar, then the value of $$p q r-(p+q+r)$$ is