2013 MI P1 Q11

Referred to the origin \(O\), the position vectors of the points \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively, and \(OAB\) is a triangle.

Show that the area of the triangle is given by \(\frac{1}{2}\left| \mathbf{a}\times \mathbf{b} \right|\).[2]

Show that the maximum area of the triangle is \(\frac{1}{2}\left| \mathbf{a} \right|\left| \mathbf{b} \right|\).[2]

By considering \(\left| \mathbf{a}\centerdot \mathbf{b} \right|\) where \(\mathbf{a}=\left( \begin{matrix} {{a}_{1}} \\ {{a}_{2}} \\ {{a}_{3}} \\ \end{matrix} \right)\) and \(\mathbf{b}=\left( \begin{matrix} {{b}_{1}} \\ {{b}_{2}} \\ {{b}_{3}} \\ \end{matrix} \right)\), show that

\({{\left( {{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+{{a}_{3}}{{b}_{3}} \right)}^{2}}\le \left( {{a}_{1}}^{2}+{{a}_{2}}^{2}+{{a}_{3}}^{2} \right)\left( {{b}_{1}}^{2}+{{b}_{2}}^{2}+{{b}_{3}}^{2} \right)\)[3]