Transformations on the plane using algorithms
Points on a number plane that can be represented by a coordinate $(x,y)$(x,y) can also be represented by the column matrix
Multiple points can be represented in the one matrix. The coordinates $(3,6)$(3,6), $(-2,4)$(−2,4) and $(4,5)$(4,5) can be represented by the matrix:
Transforming Points
Representing points by column matrices is useful when we want to move all the points in the plane in certain ways. The matrix operations of addition and multiplication give us two kinds of these transformations in the plane. We follow specific algorithms to perform these operations on matrices.
ADDITION
TRANSLATION
Starting with the point$(x,y)$(x,y) we can slide the point $a$a units in the $x$x-direction and $b$b units in the $y$y-direction by adding the column matrix as shown below.
This kind of transformation is called a translation.
Example 1
Use matrices to transform the point $(3,-4)$(3,−4) by $3$3 in the$x$x- direction.
The coordinates of the point after the transformation is $(6,-4)$(6,−4)
MULTIPLICATION
We can multiply a column matrix, representing a point, by a $2\times2$2×2 transformation matrix as follows.
The algorithm for multiplying a matrix representing $3$3points by a $2\times2$2×2transformation matrix looks like this
Depending on how the transformation matrix is chosen, we can use it to reflect points across any line, dilate vertically and horizontally by specified dilation factors and rotate about the origin.
REFLECTION
Reflections of the plane in the x-axis and y-axis respectively are given by the transformation matrices
Example 2
Use matrix multiplication to find the coordinates of the points $(3,6)$(3,6), $(-2,4)$(−2,4) and $(4,5)$(4,5) after reflection in the $x$x-axis.
The coordinates of the points after the transformation are $(3,6)$(3,6), $(-2,4)$(−2,4) and$(4,5)$(4,5)
DILATION
The following matrix multiplication will dilate (stretch or squash) the point $(x,y)$(x,y) by a factor of $a$a with respect to the $x$x-axis and by a factor of $b$b with respect to the $y$y-axis.
ROTATION
If points in the plane are to be rotated anticlockwise through an angle $\theta$θ, we see that $(1,0)\rightarrow(\cos\theta,\sin\theta)$(1,0)→(cosθ,sinθ)and $(0,1)\rightarrow(-\sin\theta,\cos\theta)$(0,1)→(−sinθ,cosθ)
Therefore, the transformation matrix is 
Worked Examples
QUESTION 1
A set of three points are given in the matrix below.
| $-3$−3 | $-1$−1 | $-7$−7 | ||||
| $-4$−4 | $-8$−8 | $-2$−2 |
They are all translated using the transformation matrix:
$0$0 $0$0 $0$0 $7$7 $7$7 $7$7 Calculate the coordinates of the newly translated points:
$-3$−3 $-1$−1 $-7$−7 $+$+ $0$0 $0$0 $0$0 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $-4$−4 $-8$−8 $-2$−2 $7$7 $7$7 $7$7 $\editable{}$ $\editable{}$ $\editable{}$
QUESTION 2
Consider the following:
Plot the following four points on the graph below:
$A\left(-2,1\right),B\left(-7,1\right),C\left(-7,3\right),D\left(-2,3\right)$A(−2,1),B(−7,1),C(−7,3),D(−2,3)
Loading Graph...State the coordinates of the four points if we translated each $10$10 units right:
$A'$A′ $=$= $\left(\editable{},\editable{}\right)$(,)
$B'$B′ $=$= $\left(\editable{},\editable{}\right)$(,)
$C'$C′ $=$= $\left(\editable{},\editable{}\right)$(,)
$D'$D′ $=$= $\left(\editable{},\editable{}\right)$(,)
We can use a simple transformation matrix to calculate the location of the new points.
Complete the values in the transformation matrix below
$-2$−2 $-7$−7 $-7$−7 $-2$−2 $+$+ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $=$= $8$8 $3$3 $3$3 $8$8 $1$1 $1$1 $3$3 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $1$1 $1$1 $3$3 $3$3
QUESTION 3
If we have three points we wish to translate $6$6 places down, complete the transformation matrix we could use.
$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
QUESTION 4
Consider the following:
By considering the two points with array:
$1$1 $0$0 $0$0 $1$1 determine the transformation matrix that reflects points over the $x$x-axis.
$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Consider the points $A,B,C$A,B,C in the graph below.
Loading Graph...Use the transformation matrix from (a) to reflect the points $A,B,C$A,B,C about the $x$x-axis.
$1$1 $0$0 $\times$× $-4$−4 $-9$−9 $-2$−2 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $0$0 $-1$−1 $9$9 $0$0 $9$9 $\editable{}$ $\editable{}$ $\editable{}$ By considering the two points with array:
$1$1 $0$0 $0$0 $1$1 determine the transformation matrix that will rotate points $90^\circ$90° anticlockwise.
$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ In part (b), we found the points $A',B',C'$A′,B′,C′. They are plotted in the graph below.
Loading Graph...Use the transformation matrix from (c) to reflect the points $A',B',C'$A′,B′,C′ drawn above.
$0$0 $-1$−1 $\times$× $-4$−4 $-9$−9 $-2$−2 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $1$1 $0$0 $-9$−9 $0$0 $-9$−9 $\editable{}$ $\editable{}$ $\editable{}$