A vector a makes equal acute angles on the coordinate axis. Then the projection of vector $$\mathbf{b}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ on $$\mathbf{a}$$ is

The diagonals of a parallelogram are the vectors $$3 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$$. and $$-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}$$. Then the length of the shorter side of parallelogram is

If $$\mathbf{a} \cdot \mathbf{b}=0$$ and $$\mathbf{a}+\mathbf{b}$$ makes an angle $$60^{\circ}$$ with $$a$$, then

If the area of the parallelogram with $$\mathbf{a}$$ and $$\mathbf{b}$$ as two adjacent sides is 15 sq units, then the area of the parallelogram having $$\mathrm{3 a+2 b}$$ and $$\mathbf{a}+3 \mathbf{b}$$ as two adjacent sides in sq units is