Worksheet: Rotation Matrices

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In this worksheet, we will practice finding the matrix of a linear transformation which rotates every vector in R² by a given angle.

Q1:

Find the matrix for the linear transformation that rotates every vector in ℝ 2 through an angle of 𝜋 3 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E − 1 0 0 − 1

Q2:

Find the matrix for the linear transformation that rotates every vector in ℝ 2 through an angle of 𝜋 4 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E − 1 0 0 − 1

Q3:

Find the matrix for the linear transformation which rotates every vector in ℝ 2 through an angle of 𝜋 1 2 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ 1 − √ 2 2 √ 2 − √ 3 2 √ 3 − √ 2 2 1 − √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ √ 2 − √ 6 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ 1 − √ 2 2 √ 3 − √ 2 2 √ 3 − √ 2 2 1 − √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 + √ 2 4 √ 2 − √ 6 4 √ 6 − √ 2 4 √ 6 + √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q4:

Find the matrix for the linear transformation which rotates every vector in ℝ 2 through an angle of 2 𝜋 3 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 − 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 √ 3 2 √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 − √ 3 2 √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q5:

Find the matrix for the linear transformation which rotates every vector in ℝ 2 through an angle of 5 𝜋 1 2 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ − 1 − √ 2 2 √ 2 − √ 3 2 √ 3 − √ 2 2 − 1 − √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 − √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ − 1 − √ 2 2 √ 3 + √ 2 2 √ 3 − √ 2 2 − 1 − √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 − √ 2 4 − √ 2 − √ 6 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 − √ 2 4 − √ 2 − √ 6 4 − √ 2 − √ 6 4 √ 2 − √ 6 4 ⎞ ⎟ ⎟ ⎟ ⎠

Q6:

Find the matrix for the linear transformation which rotates every vector in ℝ 2 through an angle of − 𝜋 3 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 − √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 − √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q7:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

  • A
  • B
  • C
  • D
  • E

Q8:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

  • A
  • B
  • C
  • D
  • E

Q9:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

  • A
  • B
  • C
  • D
  • E

Q10:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ 2 counterclockwise about the origin through an angle of 5 𝜋 1 2 .

  • A ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 − √ 2 4 − √ 6 + √ 2 4 √ 6 + √ 2 4 − √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − √ 6 − √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 − √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 + √ 2 4 − √ 6 + √ 2 4 − √ 6 + √ 2 4 √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ √ 6 − √ 2 4 − √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − √ 6 + √ 2 4 − √ 6 + √ 2 4 − √ 6 + √ 2 4 − √ 6 − √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠

Q11:

Describe the geometric effect of the transformation produced by the matrix ⎛ ⎜ ⎜ ⎜ ⎝ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠ .

  • Aa rotation through an angle of − 9 0 ∘
  • Ba rotation through an angle of − 4 5 ∘
  • Ca rotation through an angle of 9 0 ∘
  • Da rotation through an angle of 4 5 ∘
  • Ea rotation through an angle of 1 3 5 ∘

Q12:

A rotation with centre the origin sends the vector 3 4 to 4 3 . Find the matrix representation of this rotation.

  • A 0 1 1 0
  • B ⎛ ⎜ ⎜ ⎝ 4 5 3 5 − 3 5 4 5 ⎞ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ 2 4 2 5 7 2 5 7 2 5 − 2 4 2 5 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎝ 2 4 2 5 7 2 5 − 7 2 5 2 4 2 5 ⎞ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎝ 4 5 3 5 3 5 − 4 5 ⎞ ⎟ ⎟ ⎠

Q13:

Let 𝐴 be the matrix of the transformation which rotates all vectors in ℝ 2 through an angle of 𝜃 , where 0 < 𝜃 ≤ 2 𝜋 . For which values of 𝜃 does 𝐴 have a real eigenvalue?

  • A 𝜃 = 0 and 𝜃 = 3 𝜋 2
  • B 𝜃 = 0 and 𝜃 = 𝜋 2
  • C 𝜃 = 0 and 𝜃 = 𝜋 3
  • D 𝜃 = 0 and 𝜃 = 𝜋
  • E 𝜃 = 𝜋 2 and 𝜃 = 𝜋

Q14:

Consider the linear map 𝑇 ∶ ℝ → ℝ 2 2 , which acts by rotating a vector 𝜃 ∘ counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

  • A
  • B
  • C
  • D
  • E

Q15:

Consider the linear transformation , which rotates each vector 90 degrees counterclockwise about the positive -axis then 45 degrees counterclockwise about the positive -axis. Find the matrix representation of in the standard basis.

  • A
  • B
  • C
  • D

Q16:

Consider the linear transformation , which rotates each vector 45 degrees counterclockwise about the positive -axis then 90 degrees counterclockwise about the positive -axis. Find in the standard basis.

  • A
  • B
  • C
  • D

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