If $\mathbf{a}=-4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=\sqrt{2} \hat{\mathbf{i}}-\sqrt{2} \hat{\mathbf{j}}$ are two vectors, then angle between the vectors $2 \mathbf{a}$ and $\frac{\mathbf{b}}{2}$ is

A unit vector perpendicular to the vectors $a=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is

Let $|\hat{\mathbf{a}}|=2=|\hat{\mathbf{b}}|=3$ and the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ be $\frac{\pi}{3}$. If a parallelogram is constructed with adjacent sides $2 \hat{\mathbf{a}}+3 \hat{\mathbf{b}}$ and $\hat{\mathbf{a}}-\hat{\mathbf{b}}$, then its shorter diagonal is of length