(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 ...
Question 1 Report
(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 \(Z \to Z_{1}\).
Answer Details
(a)
Given
\[p\begin{pmatrix}2\\3\end{pmatrix}+q\begin{pmatrix}5\\-2\end{pmatrix}=\begin{pmatrix}-4\\13\end{pmatrix}.\]
Equating corresponding components gives
\[2p+5q=-4 \qquad (1)\]
\[3p-2q=13 \qquad (2)\]
Multiply (1) by 2 and (2) by 5:
\[4p+10q=-8\]
\[15p-10q=65\]
Adding,
\[19p=57\]
\[p=3.\]
Substituting into (1),
\[2(3)+5q=-4\]
\[5q=-10\]
\[q=-2.\]
Therefore, \(\boxed{p=3,\ q=-2}\).
(b)(i) and (ii)
Using the scale \(2\text{ cm}\) to \(2\) units on both axes, the quadrilateral and its image are plotted below. The vertices are joined in the order \(W\to X\to Y\to Z\to W\), and similarly for the image.
A rotation of \(90^\circ\) anticlockwise about the origin maps \((x,y)\) onto \((-y,x)\). Hence:
\[\begin{aligned}W(2,3)&\longmapsto W_1(-3,2),\\X(4,-1)&\longmapsto X_1(1,4),\\Y(-3,-4)&\longmapsto Y_1(4,-3),\\Z(-3,2)&\longmapsto Z_1(-2,-3).\end{aligned}\]
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