If $$\alpha=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}, \beta=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$$, then express $$\beta$$ in the form $$\beta=\beta_1+\beta_2$$ where $$\beta_1$$ is parallel to $$\alpha$$ and $$\beta_2$$ is perpendicular to $$\alpha$$, then $$\beta_1$$ is given by

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