(a) The operation (*) is defined on the set of real numbers, R, by \(x
(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 \(8 * y = 8\frac{1}{4}\), find the value of y.
(b) In \(\Delta ABC, \overline{AB} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(\overline{AC} = \begin{pmatrix} 3 \\ -8 \end{pmatrix}\). If P is the midpoint of \(\overline{AB}\), express \(\overline{CP}\) as a column vector.
Explanation
(a) (i) \(3 * \frac{2}{5} = \frac{3 + \frac{2}{5}}{2}\)
= \(\frac{\frac{17}{5}}{2}\)
= \(\frac{17}{10}\)
(ii) \(8 * y = 8\frac{1}{4}\)
\(\frac{8 + y}{2} = \frac{33}{4}\)
\(66 = 32 + 4y \implies 4y = 66 - 32 = 34\)
\(y = \frac{34}{4} = 8.5\)
(b) \(\overrightarrow{AP} = \frac{1}{2}(\overrightarrow{AB})\)
= \(\frac{1}{2} \begin{pmatrix} -4 \\ 6 \end{pmatrix}\)
= \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\)
Midpoint = \(\overrightarrow{AP} + \overrightarrow{PC} = \overrightarrow{AC}\)
\(\overrightarrow{PC} = \overrightarrow{AC} - \overrightarrow{AP}\)
= \(\begin{pmatrix} 3 \\ -8 \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)
= \(\begin{pmatrix} 5 \\ -11 \end{pmatrix}\)
\(\overrightarrow{CP} = -1 \times \begin{pmatrix} 5 \\ -11 \end{pmatrix}\)
= \(\begin{pmatrix} -5 \\ 11 \end{pmatrix}\)