The scalar product of vectors $$\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$$ and a unit vector along the sum of vectors $$\bar{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}$$ and $$\bar{c}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}$$ is one, then the value of $$\lambda$$ is

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