The magnitude of the projection of the vector $$2 \hat{i}+\hat{j}+\hat{k}$$ on the vector perpendicular to the plane containing the vectors $$\hat{i}+\hat{j}+\hat{k}$$ and $$\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$$ is

If $$\bar{a}, \bar{b}$$ and $$\bar{c}$$ are any three non-zero vectors, then $$(\bar{a}+2 \bar{b}+\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]=$$

Vectors $$\overline{\mathrm{a}}$$ and $$\overline{\mathrm{b}}$$ are such that $$|\overline{\mathrm{a}}|=1 ;|\overline{\mathrm{b}}|=4$$ and $$\bar{a} \cdot \bar{b}=2$$. If $$\bar{c}=2 \bar{a} \times \bar{b}-3 \bar{b}$$, then the angle between $$\bar{b}$$ and $$\bar{c}$$ is

Two adjacent sides of a parallelogram are $$2 \hat{i}-4 \hat{j}+5 \hat{k}$$ and $$\hat{i}-2 \hat{j}-3 \hat{k}$$, then the unit vector parallel to its diagonal is