$$\overline{\mathrm{u}}, \overline{\mathrm{v}}, \overline{\mathrm{w}}$$ 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

Let $$\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \bar{b}=\hat{i}+\hat{j}$$ and $$\bar{c}$$ be a vector such that $$|\bar{c}-\bar{a}|=4,|(\bar{a} \times \bar{b}) \times \bar{c}|=3$$ and the angle between $$\overline{\mathrm{c}}$$ and $$\overline{\mathrm{a}} \times \overline{\mathrm{b}}$$ is $$\frac{\pi}{6}$$, then $$\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}$$ is equal to