ROTATION TRANSFORMATION MATRIX

About "Rotation transformation matrix"

"Rotation transformation matrix" is the matrix which can be used to make rotation transformation of a figure.

Even though students can get this stuff on internet, they do not understand exactly what has been explained. 

To make the students to understand the stuff "Rotation transformation using matrix", we have explained the different rules which we apply to make rotation-transformation.

Rotation transformation matrix - Rule 

90° rotation (clock wise)

90° rotation (counter clock wise)

180° rotation (clock wise and counter clock wise)

Once students understand the rules which they have to apply for rotation transformation, they can easily make rotation  transformation of a figure.

Let us consider the following example to have better understanding of rotation transformation using matrices.

Question :

Let A ( -2, 1), B (2, 4) and (4, 2) be the three vertices of a triangle. If this triangle is rotated about 90° counter clockwise, find the vertices of the rotated image A'B'C' using matrices.

Solution:

Step 1 :

First we have to write the vertices of the given triangle ABC  in matrix form as given below.

Step 2 :

Since the triangle ABC is rotated about 90° counter clockwise, to get the rotated image, we have to multiply the above matrix by  the matrix given below.

Step 3 :

Now, let us multiply the two matrices.

Step 4 :

Now we  can get vertices of the rotated image A'B'C' from the resultant matrix. 

Vertices of the reflected image are

                               A' (-1 , -2) , B' (-4 , 2) and C' (-2 , 4)

After having gone through the example given above, we hope that the students would have understood the way in which they have to find the vertices of the rotated image using matrices.

How to sketch the rotated figure?

1. First we have to plot the vertices of the pre-image.

2. In the above problem, the vertices of the pre-image are

A ( -2, 1 ) , B ( 2, 4 ) and C ( 4 , 2 )

3. When we plot these points on a graph paper, we will get the figure of the pre-image (original figure).

4. When we rotate the given figure about 90° counter clock wise,  vertices of the image are

                              A' (-1 , -2) , B' (-4 , 2) and C' (-2 , 4)

7. When plot these points on the graph paper, we will get the figure of the image (rotated figure).

Rotation transformation matrix - Practice problems

Problem 1 : 

Let K (-4, -4), L (0, -4), M (0, -2) and N(-4, -2) be the vertices of a rectangle. If this rectangle is rotated 90° clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are                     

K' (-4, 4) , L' (-4, 0), M' (-2, 0) and N' (-2, 4)

GRAPH

Problem 2 : 

Let R (-3, 5), S (-3, 1), T (0, 1), U (0, 2), V (-2, 2) and W (-2, 5) be the vertices of a closed figure.If this figure is rotated 90° clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are       

R' (5, 3) , S' (1, 3), T' (1, 0), U' (2, 0), V' (2, 2) and W' (5, 2)  

GRAPH

Problem 3 : 

Let P (-1, -3), Q (3, -4), R (4, 0) and S (0, -1) be the vertices of a closed figure. If the figure is rotated 90° clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

P' (-3, 1) , Q' (-4, -3), R ( 0, -4) and S' (-1, 0)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 4 : 

Let F (-4, -2), G (-2, -2) and H (-3, 1) be the three vertices of a triangle. If this triangle is rotated 90° counterclockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

                       F' (2, -4) , G' (2, -2) and H' (-1, -3)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 5 : 

Let A (-4, 3), B (-4, 1), C (-3, 0), D (0, 2) and E (-3,4) be the vertices of a closed figure.If this figure is rotated 90° counterclockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

A' (-3, -4) , B' (-1, -4), C' (0, -3), D' (-2, 0) and E' (-4, -3)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 6 : 

Let D (-1, 2), E (-5, -1) and F (1, -1) be the vertices of a triangle.If the triangle is rotated 90° counterclockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

D' (-2, -1) , E' (1, -5) and F' (1, 1) 

Let us look at the next problem on "Rotation transformation matrix"

Problem 7 : 

Let P (-2, -2), Q (1, -2) R (2, -4) and S (-3, -4) be the vertices of a four sided closed figure. If this figure is rotated 180° clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are                     

                       P' (2, 2) , Q' (-1, 2), R' (-2, 4) and S' (3, 4)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 8 : 

Let K (1, 4), L (-1, 2), M (1, -2) and N (3, 2) be the vertices of a four sided closed figure.If this figure is rotated 180° counter clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are          

K' (-1, -4) , L' (1, -2), M' (-1, 2) and N' (-3, -2)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 9 : 

Let T (1, -3), U (5, -5), V (3, -3) and W (5, -1) be the vertices of a closed figure.If this figure is rotated 270° counter clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

T' (-3, -1) , U' (-5, -5), V' (-3, -3) and  W' (-1, -5)  

Let us look at the next problem on "Rotation transformation matrix"

Problem 10 : 

Let A (-5, 3), B (-4, 1), C (-2, 1) D (-1, 3) and E (-3,  4) be the vertices of a closed figure.If this figure is rotated 270° clockwise, find the vertices of the rotated figure and graph.

Solution : 

Vertices of the rotated figure are        

    A' (-3, -5) , B' (-1, -4), C' (-1, -2), D' (-3, -1) and E' (-4, -3)  

We hope that the students would have understood the stuff given on "Rotation transformation matrix"

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