Referred to the origin \(O\), points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(\mathbf{a}+\mathbf{b}-\mathbf{c}\) and \(\mathbf{a}-\mathbf{b}+\mathbf{c}\) are parallel.
Using vector product, show that \(\overrightarrow{OA}\) and \(\overrightarrow{BC}\) are parallel.[4]
It is now given that \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors, \(\overrightarrow{BC}=2\overrightarrow{OA}\) and angle \(OBC=60^\circ\).
Using scalar product, show that \(\overrightarrow{OB}\) and \(\overrightarrow{OC}\) are perpendicular.[3]
The point \(H\) has position vector \(\mathbf{h}\). Given that \(\mathbf{h}=\mathbf{b}\times\mathbf{c}\) and \(|\mathbf{c}|=\sqrt3\), find the exact volume of the pyramid \(HOBC\).[3]
[The volume of a pyramid is \(\dfrac13\times\) base area \(\times\) height.]