(a) If (x = begin{pmatrix} 2 \ 3 end{pmatrix}, y = begin{pmatrix} 5 \ -2 end{pmatrix}) and (z = begin{pmatrix} -4 \ 13 end{pmatrix}), find the scalars p and q such that (px + qy = z).
(b)(i) Using the scale of 2cm to 2 units on both axis, draw on a graph paper two perpendicular axis x and y for (-5 leq x leq 5, -5 leq y leq 5) respectively.
(ii) Draw, on the graph paper, indicating clearly the vertices and their coordinates,
(1) the quadrilateral WXYZ with W(2, 3), X(4, -1), Y(-3, -4) and Z(-3, 2).
(2) the image (W_{1}X_{1}Y_{1}Z_{1}) of the quadrilateral WXYZ under an anti-clockwise rotation of 90° about the origin where (W to W_{1}, X to X_{1}, Y to Y_{1}) and (Z to Z_{1}).
Explanation
(a) px + qy = z
(p begin{pmatrix} 2 \ 3 end{pmatrix} + q begin{pmatrix} 5 \ -2 end{pmatrix} = begin{pmatrix} -4 \ 13 end{pmatrix})
(begin{pmatrix} 2p + 5q \ 3p – 2q end{pmatrix} = begin{pmatrix} -4 \ 13 end{pmatrix})
Thus,
(2p + 5q = -4 …..(1))
(3p – 2q = 13 ……(2))
Solving the equations, we have p = 3 and q = -2.
(b)(i)
(ii) See attached graph for the quadrilateral W(2, 3), X(4, -1), Y(-3, -4) and Z(-3,2).
(iii) See attached for the image for the quadrilateral WXYZ under an anticlockwise rotation of 90° is (W_{1}(-3, 2), X_{1}(1, 4), Y_{1}(4, -3)) and (Z_{1}(-2, -3)).