If $$|\mathbf{a}|=2$$ and $$|\mathbf{b}|=3$$ and the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is $$120^{\circ}$$, then the length of the vector $$\left|\frac{\mathbf{a}}{2}-\frac{\mathbf{b}}{3}\right|$$ is

If $$|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=36$$ and $$|\mathbf{a}|=3$$, then $$|\mathbf{a}|$$ is equal to

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