2016 TJC P1 Q4

Relative to the origin \(O\), the points \(A\), \(B\), \(M\) and \(N\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{m}\) and \(\mathbf{n}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors. It is given that \(\mathbf{m}=\lambda \mathbf{a}+\left( 1-\lambda \right)\mathbf{b}\) and \(\mathbf{n}=2\left( 1-\lambda \right)\mathbf{a}-\lambda \mathbf{b}\) where \(\lambda\) is a real parameter.

Show that \(\mathbf{m}\times \mathbf{n}=\left( 3{{\lambda }^{2}}-4\lambda +2 \right)\left( \mathbf{b}\times \mathbf{a} \right)\).[2]

It is given that \(\left| \mathbf{a} \right|=3\), \(\left| \mathbf{b} \right|=4\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\)is \(\frac{\pi }{6}\). Find the smallest area of the triangle \(MON\) as \(\lambda\) varies.[4]