Worksheet: Rotation Matrices

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 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 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 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 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 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 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 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 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 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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