(a) The operation (*) is defined on the set of real numbers, R, by \(x * y = \frac{x + y}{2}, x, y \in R\). (i) Evaluate \(3 * \frac{2}{5}\). (ii) If \(8 * ...

Answer Details

(a) The operation is \(x * y = \dfrac{x + y}{2}\).

(i) \(3 * \dfrac{2}{5} = \dfrac{3 + \tfrac{2}{5}}{2} = \dfrac{\tfrac{17}{5}}{2} = \dfrac{17}{10} = 1\tfrac{7}{10}\).

(ii) \(8 * y = 8\tfrac{1}{4}\) means \(\dfrac{8 + y}{2} = \dfrac{33}{4}\).

\[8 + y = \frac{33}{2} = 16.5 \Rightarrow y = 8.5 = 8\tfrac{1}{2}.\]

(b) Take \(A\) as the reference point. Then \(\overline{AB} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(\overline{AC} = \begin{pmatrix} 3 \\ -8 \end{pmatrix}\).

\(P\) is the midpoint of \(\overline{AB}\), so

\[\overline{AP} = \frac{1}{2}\overline{AB} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}.\]

Then

\[\overline{CP} = \overline{AP} - \overline{AC} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} - \begin{pmatrix} 3 \\ -8 \end{pmatrix} = \begin{pmatrix} -5 \\ 11 \end{pmatrix}.\]

Therefore \(\overline{CP} = \begin{pmatrix} -5 \\ 11 \end{pmatrix}\).