FP1 Matrices Transformations BAT use matrix products to represent combinations of transformations

WB 11a The points A(1,0), B(0,1) and C(2,0) ate the vertices of a triangle T. The triangle T is rotated 90° anticlockwise around (0,0) and then the image T’ is reflected in the line y = x to obtain the triangle T’’. a) On separate diagrams, draw T, T’ and T’’ b) i) Find the matrix P such that P(T) = T’ ii) Find the matrix Q such that Q(T’) = T’’ c) By finding a matrix product, find the single matrix that will perform a 90° anticlockwise rotation followed by a reflection in y = x (0,1) (1,0) (-1,0) First transformation, the 90° anticlockwise rotation T A B C Rotated 90° anticlockwise about (0,0) T’ A C B Reflected in y = x T’’ A C B 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑝𝑎𝑖𝑟: 1 0 0 1 Replace the coordinates! 𝑁𝑒𝑤 𝑝𝑎𝑖𝑟: 0 −1 1 0 𝑷= 0 −1 1 0

WB 11b The points A(1,0), B(0,1) and C(2,0) ate the vertices of a triangle T. The triangle T is rotated 90° anticlockwise around (0,0) and then the image T’ is reflected in the line y = x to obtain the triangle T’’. a) On separate diagrams, draw T, T’ and T’’ b) i) Find the matrix P such that P(T) = T’ ii) Find the matrix Q such that Q(T’) = T’’ c) By finding a matrix product, find the single matrix that will perform a 90° anticlockwise rotation followed by a reflection in y = x (0,1) (1,0) 2nd transformation, the reflection in y = x (remember use the standard starting coordinates) T A B C Rotated 90° anticlockwise about (0,0) T’ A C B Reflected in y = x T’’ A C B 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑝𝑎𝑖𝑟: 1 0 0 1 Replace the coordinates! 𝑁𝑒𝑤 𝑝𝑎𝑖𝑟: 0 1 1 0 𝑷= 0 −1 1 0 𝑸= 0 1 1 0

Rotated 90° anticlockwise about (0,0) WB 11c The points A(1,0), B(0,1) and C(2,0) ate the vertices of a triangle T. The triangle T is rotated 90° anticlockwise around (0,0) and then the image T’ is reflected in the line y = x to obtain the triangle T’’. a) On separate diagrams, draw T, T’ and T’’ b) i) Find the matrix P such that P(T) = T’ ii) Find the matrix Q such that Q(T’) = T’’ c) By finding a matrix product, find the single matrix that will perform a 90° anticlockwise rotation followed by a reflection in y = x T A B C Rotated 90° anticlockwise about (0,0) T’ A C B Reflected in y = x T’’ A C B 𝑷= 0 −1 1 0 𝑸= 0 1 1 0 It is important to note that P acts first, and then Q acts on P This is written as … 𝑸𝑷= 0 1 1 0 0 −1 1 0 𝑸𝑷= 1 0 0 −1 Consider how the original coordinates have moved This is the same as a reflection in the x-axis!

This is a 90° rotation clockwise around (0,0) WB 12 The following matrices represent three different transformations: 𝑷= 1 1 2 3 𝑸= 1 2 0 1 𝑹= 3 7 −1 −2 Find the matrix representing the transformation represented by R, followed by Q, followed by P and give a geometrical interpretation of this transformation. Ensure you get the order correct! R acts first, is acted on by Q which is then acted on by P PQR = (PQ)R PQ = 1 1 2 3 1 2 0 1 = 1 3 2 7 (PQ)R = 1 3 2 7 3 7 −1 −2 = 0 1 −1 0 This is a 90° rotation clockwise around (0,0) (0,1) (1,0) (1,0) (0,-1)

END