Transformations

Questions

• 1. What matrix will have he effect of a stretch factor 7 in the x direction?
• 2. What matrix will have he effect of a reflection in y=x?
• 3. What matrix will have he effect of a 90 degree ACW rotation about the origin?
• 4. What matrix will have he effect of a enlargement with a scale factor of 5?
• 5. What matrix will have he effect of a reflection in the y axis?
• 6. What matrix will have he effect of a 180 degree rotation about the origin?
• 7. What are the coordinates of the matrix [2 0][-1 3] and what shape does it make?
• 8. What matrix has the effect of a 120 degree AC rotation about the origin?
• 9. What matrix reflects in the line y=-2x
• 10. For the matrix 1/root(5)[1 2][2 -1] find the line of reflection

Answers 1-6...

• 1. What matrix will have he effect of a stretch factor 7 in the x direction? , [7 0][0 1]
• 2. What matrix will have he effect of a reflection in y=x? , [0 1][1 0]
• 3. What matrix will have he effect of a 90 degree ACW rotation about the origin? , [0 -1][1 0]
• 4. What matrix will have he effect of a enlargement with a scale factor of 5? , [5 0][0 5]
• 5. What matrix will have he effect of a reflection in the y axis? , [-1 0][0 1]
• 6. What matrix will have he effect of a 180 degree rotation about the origin? , [-1 0][0 -1]

7....

• 7. What are the coordinates of the matrix [2 0][-1 3] and what shape does it make?
• [1 0] -> [2 0] , [0 1] -> [-1 3] , [1 1] -> [1 3] , [0 0] -> [0 0]
• Coordinates: (2,0) , (-1,3) , (1,3) , (0,0)
• Shape = Parallelogram

Questions 8 and 9

• 8. What matrix has the effect of a 120 degree AC rotation about the origin?
• [cosθ -sinθ][sinθ cosθ] , so [cos120 -sin120][sin120 cos120]
• cos120 = -0.5 , sin120 = root(3)/2
• [-0.5 -root(3)/2][root(3)/2 -0.5]
• 9. What matrix reflects in the line y=-2x
• tanθ=-2 , so θ=tan-1(-2)
• tan-1(-2) x2 (needed for cos/sin2θ) , then cos(126.869...) , sin(126.869...) =
• [-0.6 -0.0104...][-0.0104 0.6]

10...

• 10. For the matrix 1/root(5)[1 2][2 -1] find the line of reflection
• Expand to [1/root(5) 2/root(5)][2/root(5) -1/root(5)]
• Now we know cos2θ = 1/root(5) and sin2θ=2/root(5) , so we cos-1/sin-1 these values and then divide by 2 to get our thetas.
• cos-1(1/root(5)) = 63.43 , 63.43 = 2θ , so θ = 31.71 , sin-1(2/root(5)) = 63.43 , 63.43 = 2θ , so θ = 31.71
• Now we know both thetas = 31.71, so tanθ = 31.43. We can now do tan(31.43) giving 0.618.
• The final answer is y=0.618