Let \(a, b, c\) be vectors such that \(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \begin{pmatrix} -3 \\ -12 \\ 4 \end{pmatrix} .\) Find the area of the triangle whose vertices are \(a, b\) , and \(c.\)
The area of the triangle formed by \(a, b,\) and \(c\) is given by
\(\begin{align*} \frac{1}{2} \|(\mathbf{b} - \mathbf{a}) \times (\mathbf{c} - \mathbf{a})\| &= \frac{1}{2} \|\mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} - \mathbf{a} \times \mathbf{c} + \mathbf{a} \times \mathbf{a}\| \\ &= \frac{1}{2} \|\mathbf{b} \times \mathbf{c} + \mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{a}\| \\ &= \frac{1}{2} \left\| \begin{pmatrix} -3 \\ -12 \\ 4 \end{pmatrix} \right\| \\ \end{align*}\)
which gives you 13/2 as your final answer.
