If the unit vectors $\mathbf{a}$ and $\mathbf{b}$ are inclined at $2 \theta$ and $|\mathbf{a}-\mathbf{b}|<1$, then if $0<\theta<\pi, \theta$ lies in the interval.

The volume of the parallelopiped whose edges are represented by $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is

A unit vector perpendicular to both the vectors $$\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+\hat{\mathbf{k}}$$ is

Let $$a, b$$ and $$c$$ be three unit vectors such that $$a \times(b \times c)=\frac{\sqrt{3}}{2}(b+c)$$. If $$b$$ is not parallel to $$c$$, then the angle between $$a$$ and $$b$$ is