Given AB = -4 7 and CA = 3 −5 .

Here are the solutions to the problems.

12. (a) (i) Calculate $\vec{BC}$
Given $\vec{AB} = \begin{pmatrix} -4 \\ 7 \end{pmatrix}$ and $\vec{CA} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}$.

Step 1: Express $\vec{BC}$ in terms of the given vectors.
We use the property of vector addition: $\vec{BC} = \vec{BA} + \vec{AC}$.

Step 2: Convert the given vectors to the required form.
$\vec{BA} = -\vec{AB} = -\begin{pmatrix} -4 \ 7 \end{pmatrix} = \begin{pmatrix}