J20

Consider a reflection of the unit square in the x-axis. Point I(1, 0) is mapped
Onto I 1(1, 0). Point J(0, 1) is mapped onto J 1(0, -1). Point O(0, 0) is mapped
onto O 1(0, 0) and point K(1, 1) is mapped onto K 1(-1, -1).
1
0
1
0
The position vectors of I 1 = and J1 = form the matrix
0
1
0
1
1
0
When the position vectors of the unit square are premultiplied by we
0
1
get,
O1 I1
J1
K1
1
0
0
1 0 1
= 0
1 0 1
0
1
0
0 1 1
0
0 1
1
The result is the matrix formed by the position vectors of the images of the unit
1
0
square under reflection in the x-axis. This means that represents a
0
1
reflection in the x-axis.
In order to determine a matrix of transformation, we need to know the kind of
transformation it is, find the images of points I and J under that transformation, and then
write their position vectors in matrix form.
Example
A triangle with vertices P(2, 2), Q(5, 1) and R(5, 6) is rotated through 90 0 anti-clockwise
about the origin. Find the transformation matrix and the coordinates of its image.
Solution
The images of I and J under this transformation are I 1 (0,1) and J 1 (-1, 0). Therefore, the
0
1
matrix of transformation is .
1
0
When we premultiply the matrix resulting from the position vectors of points P, Q and R
by the matrix of transformation, we get: