Worksheet: Linear Transformations in Planes: Reflection

In this worksheet, we will practice finding the matrix of linear transformation of reflection along the x- or y-axis or the line of a given equation and the image of a vector under the reflection.

Q1:

Which of the following are necessary and sufficient conditions on 𝑎, 𝑏, 𝑐, and 𝑑 for the matrix ī”đ‘Žđ‘đ‘đ‘‘ī  to represent a reflection?

  • A 𝑏 = − 𝑐 , 𝑑 = − 𝑎 , 𝑎 − 𝑐 = 1 īŠ¨ īŠ¨
  • B 𝑏 = 𝑐 , 𝑑 = 𝑎 , 𝑎 + 𝑐 = 1 īŠ¨ īŠ¨
  • C 𝑏 = 𝑐 , 𝑑 = − 𝑎 , 𝑎 − 𝑐 = 1 īŠ¨ īŠ¨
  • D 𝑏 = 𝑐 , 𝑑 = − 𝑎 , 𝑎 + 𝑐 = 1 īŠ¨ īŠ¨
  • E 𝑏 = − 𝑐 , 𝑑 = − 𝑎 , 𝑎 + 𝑐 = 1 īŠ¨ īŠ¨

Q2:

A reflection in a line through the origin sends the vector ī”34ī  to ī”43ī . Find the matrix representation of this reflection.

  • A ⎡ âŽĸ âŽĸ âŽŖ 2 4 2 5 7 2 5 − 7 2 5 2 4 2 5 ⎤ âŽĨ âŽĨ âŽĻ
  • B ī” 1 1 0 1 ī 
  • C ī” 0 1 1 0 ī 
  • D ⎡ âŽĸ âŽĸ âŽŖ 4 5 3 5 − 3 5 4 5 ⎤ âŽĨ âŽĨ âŽĻ
  • E ⎡ âŽĸ âŽĸ âŽŖ 2 4 2 5 7 2 5 7 2 5 − 2 4 2 5 ⎤ âŽĨ âŽĨ âŽĻ

Q3:

A reflection in a line through the origin sends the vector ī”34ī  to ī”4−3ī . Find the matrix representation of this reflection.

  • A ⎡ âŽĸ âŽĸ âŽŖ 2 4 2 5 7 2 5 − 7 2 5 2 4 2 5 ⎤ âŽĨ âŽĨ âŽĻ
  • B ⎡ âŽĸ âŽĸ âŽŖ 4 5 3 5 3 5 − 4 5 ⎤ âŽĨ âŽĨ âŽĻ
  • C ⎡ âŽĸ âŽĸ âŽŖ 4 5 3 5 − 3 5 4 5 ⎤ âŽĨ âŽĨ âŽĻ
  • D ī” 0 1 1 0 ī 
  • E ⎡ âŽĸ âŽĸ âŽŖ 2 4 2 5 7 2 5 7 2 5 − 2 4 2 5 ⎤ âŽĨ âŽĨ âŽĻ

Q4:

Consider the reflection in the line đ‘Ļ=12đ‘Ĩ.

Find the matrix that represents this transformation.

  • A ⎡ âŽĸ âŽĸ âŽŖ 3 5 4 5 4 5 3 5 ⎤ âŽĨ âŽĨ âŽĻ
  • B ⎡ âŽĸ âŽĸ âŽŖ 1 2 − 1 2 1 2 1 2 ⎤ âŽĨ âŽĨ âŽĻ
  • C ⎡ âŽĸ âŽĸ âŽŖ 3 5 4 5 4 5 − 3 5 ⎤ âŽĨ âŽĨ âŽĻ
  • D ⎡ âŽĸ âŽĸ âŽŖ 4 5 3 5 3 5 − 4 5 ⎤ âŽĨ âŽĨ âŽĻ
  • E ⎡ âŽĸ âŽĸ âŽŖ 1 2 1 2 1 2 − 1 2 ⎤ âŽĨ âŽĨ âŽĻ

What is the image of the point (12,5) under this reflection?

  • A ī€ŧ 1 7 2 , 7 2 īˆ
  • B ī€ŧ 5 6 5 , 3 3 5 īˆ
  • C ī€ŧ 7 2 , 1 7 2 īˆ
  • D ī€ŧ 5 6 5 , 6 3 5 īˆ
  • E ī€ŧ 6 3 5 , 1 6 5 īˆ

Q5:

Consider the linear transformation which maps a point to its reflection in the đ‘Ĩ-axis.

Find the matrix 𝐴 which represents this transformation.

  • A 𝐴 = ī” 1 0 − 1 1 ī 
  • B 𝐴 = ī” 1 0 0 1 ī 
  • C 𝐴 = ī” − 1 0 1 − 1 ī 
  • D 𝐴 = ī” 1 1 0 1 ī 
  • E 𝐴 = ī” 1 0 0 − 1 ī 

Where does this transformation map the point (2,−3)?

  • A ( − 2 , 3 )
  • B ( 2 , 3 )
  • C ( − 2 , 3 )
  • D ( − 2 , − 3 )
  • E ( 2 , − 3 )

Q6:

Consider the given figure.

The points 𝑂(0,0), 𝐴(1,0), đĩ(1,1), and đļ(0,1) are corners of the unit square. This square is reflected in the line 𝑂𝐷 with equation đ‘Ļ=12đ‘Ĩ to form the image 𝑂𝐴đĩđļ∗∗∗.

As 𝐴∗ is the image of 𝐴 in the line through 𝑂 and 𝐷, 𝑚∠𝐴𝑂𝐴=2𝑚∠𝐷𝑂𝐴∗. Use this fact and the identity tantantan2𝜃=2𝜃1−𝜃īŠ¨ to find the gradient and hence equation of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đ´âˆ— from the gradient of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đˇ.

  • A đ‘Ļ = 4 3 đ‘Ĩ
  • B đ‘Ļ = − 2 3 đ‘Ĩ
  • C đ‘Ļ = 3 4 đ‘Ĩ
  • D đ‘Ļ = − 4 3 đ‘Ĩ
  • E đ‘Ļ = 2 3 đ‘Ĩ

Using the fact that ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đļ∗ is perpendicular to ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đ´âˆ—, find the equation of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đļ∗.

  • A đ‘Ļ = − 4 3 đ‘Ĩ
  • B đ‘Ļ = − 3 2 đ‘Ĩ
  • C đ‘Ļ = 4 3 đ‘Ĩ
  • D đ‘Ļ = 3 4 đ‘Ĩ
  • E đ‘Ļ = − 3 4 đ‘Ĩ

Using the fact that 𝑂đļ=𝑂𝐴=1∗∗, find the coordinates of đļ∗ and 𝐴∗.

  • A đļ = ī€ŧ 1 6 2 5 , − 9 2 5 īˆ ∗ , 𝐴 = ī€ŧ 9 2 5 , 1 6 2 5 īˆ ∗
  • B đļ = ī€ŧ − 3 5 , − 4 5 īˆ ∗ , 𝐴 = ī€ŧ 4 5 , 3 5 īˆ ∗
  • C đļ = ī€ŧ 4 5 , − 3 5 īˆ ∗ , 𝐴 = ī€ŧ 3 5 , 4 5 īˆ ∗
  • D đļ = ī€ŧ 4 7 , − 3 7 īˆ ∗ , 𝐴 = ī€ŧ 3 7 , 4 7 īˆ ∗
  • E đļ = ī€ŧ − 3 7 , 4 7 īˆ ∗ , 𝐴 = ī€ŧ 4 7 , 3 7 īˆ ∗

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line đ‘Ļ=12đ‘Ĩ.

  • A ⎡ âŽĸ âŽĸ âŽŖ 4 5 3 5 3 5 − 4 5 ⎤ âŽĨ âŽĨ âŽĻ
  • B ⎡ âŽĸ âŽĸ âŽŖ 4 7 3 7 3 7 − 4 7 ⎤ âŽĨ âŽĨ âŽĻ
  • C ⎡ âŽĸ âŽĸ âŽŖ 9 2 5 1 6 2 5 1 6 2 5 − 9 2 5 ⎤ âŽĨ âŽĨ âŽĻ
  • D ⎡ âŽĸ âŽĸ âŽŖ 3 5 4 5 4 5 − 3 5 ⎤ âŽĨ âŽĨ âŽĻ
  • E ⎡ âŽĸ âŽĸ âŽŖ 3 7 4 7 4 7 − 3 7 ⎤ âŽĨ âŽĨ âŽĻ

Q7:

Consider the matrix 𝑀=ī“đ›ŧđ›ŧđ›ŧ−đ›ŧīŸ where đ›ŧ=√22.

Find 𝑀īŠ¨.

  • A ī” 1 0 0 − 1 ī 
  • B ī” 1 0 0 1 ī 
  • C ī” − 1 0 0 1 ī 
  • D ī” 1 1 1 1 ī 
  • E ⎡ âŽĸ âŽĸ âŽŖ 1 2 1 2 1 2 1 2 ⎤ âŽĨ âŽĨ âŽĻ

Find det(𝑀).

  • A2
  • B1
  • C 1 2
  • D0
  • E − 1

By drawing the image of the unit square under the transformation, identify the geometrical transformation this matrix corresponds to.

  • Aa projection onto the line đ‘Ļ=đ‘Ĩ
  • Ba rotation by 45∘ clockwise about the point (1,0)
  • Ca reflection in the line đ‘Ļ=(22.5)đ‘Ĩtan∘
  • Da rotation of 45∘ clockwise about the origin
  • Ea reflection in the line đ‘Ļ=đ‘Ĩ

Q8:

Consider the given figure.

The points 𝑂(0,0), 𝐴(1,0), đĩ(1,1), and đļ(0,1) are corners of the unit square. This square is reflected in the line 𝑂𝐷 with equation đ‘Ļ=𝑘đ‘Ĩ to form the image 𝑂𝐴đĩđļ∗∗∗.

As 𝐴∗ is the image of 𝐴 in the line through 𝑂 and 𝐷, 𝑚∠𝐴𝑂𝐴=2𝑚∠𝐷𝑂𝐴∗. Use this fact and the identity tantantan2𝜃=2𝜃1−𝜃īŠ¨ to find the gradient and hence equation of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đ´âˆ— from the gradient of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đˇ.

  • A đ‘Ļ = 2 𝑘 1 − 𝑘 đ‘Ĩ īŠ¨
  • B đ‘Ļ = 2 𝑘 1 + 𝑘 đ‘Ĩ īŠ¨
  • C đ‘Ļ = 2 𝑘 𝑘 − 1 đ‘Ĩ īŠ¨
  • D đ‘Ļ = 𝑘 𝑘 − 1 đ‘Ĩ īŠ¨
  • E đ‘Ļ = 𝑘 1 − 𝑘 đ‘Ĩ īŠ¨

Using the fact that ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đļ∗ is perpendicular to ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đ´âˆ—, find the equation of ⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‚đļ∗.

  • A đ‘Ļ = 𝑘 − 1 2 𝑘 đ‘Ĩ īŠ¨
  • B đ‘Ļ = 1 − 𝑘 2 𝑘 đ‘Ĩ īŠ¨
  • C đ‘Ļ = 2 𝑘 1 − 𝑘 đ‘Ĩ īŠ¨
  • D đ‘Ļ = 𝑘 − 1 2 𝑘 đ‘Ĩ īŠ¨
  • E đ‘Ļ = 2 𝑘 𝑘 − 1 đ‘Ĩ īŠ¨

Using the fact that 𝑂đļ=𝑂𝐴=1∗∗, find the coordinates of đļ∗ and 𝐴∗.

  • A đļ = ī€ž 2 𝑘 1 + 𝑘 , 𝑘 − 1 1 + 𝑘 īŠ ∗ īŠ¨ , 𝐴 = ī€ž 1 − 𝑘 1 + 𝑘 , 2 𝑘 1 + 𝑘 īŠ ∗ īŠ¨
  • B đļ = ī€ž 𝑘 − 1 1 + 𝑘 , 2 𝑘 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨ , 𝐴 = ī€ž 2 𝑘 1 + 𝑘 , 1 − 𝑘 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨
  • C đļ = ī€ž 𝑘 1 + 𝑘 , 𝑘 − 1 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨ , 𝐴 = ī€ž 1 − 𝑘 1 + 𝑘 , 𝑘 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨
  • D đļ = ī€ž 2 𝑘 1 + 𝑘 , 𝑘 − 1 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨ , 𝐴 = ī€ž 1 − 𝑘 1 + 𝑘 , 2 𝑘 1 + 𝑘 īŠ ∗ īŠ¨ īŠ¨ īŠ¨
  • E đļ = ī€ž 𝑘 − 1 1 + 𝑘 , 2 𝑘 1 + 𝑘 īŠ ∗ īŠ¨ , 𝐴 = ī€ž 2 𝑘 1 + 𝑘 , 1 − 𝑘 1 + 𝑘 īŠ ∗ īŠ¨

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line đ‘Ļ=𝑘đ‘Ĩ.

  • A ⎡ âŽĸ âŽĸ âŽĸ âŽŖ 1 − 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 𝑘 − 1 1 + 𝑘 ⎤ âŽĨ âŽĨ âŽĨ âŽĻ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨
  • B ⎡ âŽĸ âŽĸ âŽĸ âŽŖ 1 − 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 𝑘 − 1 1 + 𝑘 ⎤ âŽĨ âŽĨ âŽĨ âŽĻ īŠ¨ īŠ¨
  • C ⎡ âŽĸ âŽĸ âŽĸ âŽŖ 1 − 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 1 − 𝑘 1 + 𝑘 ⎤ âŽĨ âŽĨ âŽĨ âŽĻ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨
  • D ⎡ âŽĸ âŽĸ âŽĸ âŽŖ 1 + 𝑘 1 − 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 𝑘 − 1 ⎤ âŽĨ âŽĨ âŽĨ âŽĻ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨
  • E ⎡ âŽĸ âŽĸ âŽĸ âŽŖ 1 − 𝑘 1 + 𝑘 2 𝑘 1 + 𝑘 − 2 𝑘 1 + 𝑘 1 − 𝑘 1 + 𝑘 ⎤ âŽĨ âŽĨ âŽĨ âŽĻ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨ īŠ¨

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