If $$\hat{\mathrm{a}}$$ and $$\hat{\mathrm{b}}$$ are unit vectors and $$\overline{\mathrm{c}}=\hat{\mathrm{b}}-(\hat{\mathrm{a}} \times \overline{\mathrm{c}})$$, then minimum value of $$[\hat{a} \hat{b} \bar{c}]$$ is

If $$\bar{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}$$ and $$\bar{b}=\hat{i}-\hat{j}-\hat{k}$$, then the projection of $$\bar{b}$$ in the direction of $$\bar{a}$$ is

A vector $$\bar{a}$$ has components 1 and $$2 p$$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about origin in the counter clock wise sense. If, with respect to the new system, $$\bar{a}$$ has components 1 and $$(p+1)$$, then

If $$\theta$$ is angle between the vectors $$\bar{a}$$ and $$\bar{b}$$ where $$|\bar{a}|=4,|\bar{b}|=3$$ and $$\theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$$, then $$|(\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})|^2+4(\bar{a} \cdot \bar{b})^2$$ has the value