Theory question 112
Vectors and Transformations(a) If x = (2 / 3), y = (5 / −2) and z = (−4 / 13), find scalars p and q such that px + qy = z.
(b)(i) Using a scale of 2 cm to 2 units on both axes, draw on a graph paper two perpendicular axes Ox and Oy for −5 ≤ x ≤ 5, −5 ≤ y ≤ 5 respectively. (ii) Draw, on the graph paper, indicating clearly the vertices and their coordinates, (I) the quadrilateral WXYZ with W(2, 3), X(4, −1), Y(−3, −4) and Z(−3, 2); (II) the image W1X1Y1Z1 of the quadrilateral WXYZ under an anticlockwise rotation of 90° about the origin where W→W1, X→X1, Y→Y1 and Z→Z1.
Show the model answer
(a) 2p + 5q = −4 and 3p − 2q = 13
Solving: 19p = 57 → p = 3, q = −2
(b)(i) Draw and label the axes with the given scale.
(ii)(I) Plot W(2, 3), X(4, −1), Y(−3, −4), Z(−3, 2) and join in order.
(II) Rotation of 90° anticlockwise: (x, y) → (−y, x)
W₁(−3, 2), X₁(1, 4), Y₁(4, −3), Z₁(−2, −3); plot and join.
Column vectors are written (top / bottom).