Challenging +1.2 This is a Further Maths FP3 vector proof requiring students to manipulate cross products algebraically using distributive properties and the anti-commutative property. While it involves multiple steps and vector identities, the approach is guided ('by considering AB × AC') and follows a standard proof technique. The algebraic manipulation is moderately challenging but well within reach for FM students who have studied cross products.
Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. By considering \(\overrightarrow{AB} \times \overrightarrow{AC}\), prove that the area of \(\triangle ABC\) can be expressed in the form \(\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\). [5]
Referred to a fixed origin $O$, the position vectors of three non-collinear points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. By considering $\overrightarrow{AB} \times \overrightarrow{AC}$, prove that the area of $\triangle ABC$ can be expressed in the form $\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|$. [5]
\hfill \mbox{\textit{Edexcel FP3 Q17 [5]}}