(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 = 8frac{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 = 8frac{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})