2017 HCI P1 Q5

Given that \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are non-zero vectors such that \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c}) = \mathbf{b} \times \mathbf{c}\), and \(\mathbf{b} \neq \mathbf{c}\), find the relationship between \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).[4]

It is given instead that \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) satisfy the equation \(\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}\) with \(|\mathbf{a}| = 2\), \(|\mathbf{b}| = 3\) and \(|\mathbf{c}| = 4\). Find the value of

\[\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}\][3]