Let $$\overline{\mathrm{A}}$$ be a vector parallel to line of intersection of planes $$P_1$$ and $$P_2$$ through origin. $$P_1$$ is parallel to the vectors $$2 \hat{j}+3 \hat{k}$$ and $$4 \hat{j}-3 \hat{k}$$ and $$P_2$$ is parallel to $$\hat{j}-\hat{k}$$ and $$3 \hat{i}+3 \hat{j}$$, then the angle between $$\bar{A}$$ and $$2 \hat{i}+\hat{j}-2 \hat{k}$$ is

$$\overline{\mathrm{u}}, \overline{\mathrm{v}}, \overline{\mathrm{w}}$$ are three vectors such that $$|\overline{\mathrm{u}}|=1, |\bar{v}|=2,|\bar{w}|=3$$. If the projection of $$\bar{v}$$ along $$\bar{u}$$ is equal to projection of $$\bar{w}$$ along $$\bar{u}$$ and $$\bar{v}, \bar{w}$$ are perpendicular to each other, then $$|\bar{u}-\bar{v}+\bar{w}|=$$

If $$\bar{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \bar{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}, \bar{c}=2 \hat{i}-\hat{j}+4 \hat{k}$$, then a vector $$\overline{\mathrm{d}}$$ which is parallel to vector $$\overline{\mathrm{a}} \times \overline{\mathrm{b}}$$ and which $$\overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=15$$, is

The unit vector perpendicular to each of the vectors $$\bar{a}+\bar{b}$$ and $$\bar{a}-\bar{b}$$, where $$\bar{a}=\hat{i}+\hat{j}+\hat{k}$$ and $$\overline{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$$ is