2010 NYJC P1 Q5

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, and are such that \(OACB\) is a parallelogram in an anti-clockwise sense.

1. Express \(c\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[1]
2. Show that the area of parallelogram \(OACB\) is given by \(\left| \mathbf{a}\times \mathbf{b} \right|\).[2]
3. Show that the maximum area of parallelogram \(OACB\) is \(\left| \mathbf{a} \right|\left| \mathbf{b} \right|\).[2]
4. By considering \(\left| \mathbf{a}\cdot \mathbf{b} \right|\) 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]