Worksheet: Linear Transformations in Planes: Rotation

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In this worksheet, we will practice finding the matrix of linear transformation of rotation at a given angle and the image of a vector under a given rotation linear transformation.

Q1:

Find the matrix for the linear transformation that rotates every vector in ℝ through an angle of 𝜋3.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C − 1 0 0 − 1
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 1 2 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q2:

Find the matrix for the linear transformation that rotates every vector in ℝ through an angle of 𝜋4.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B − 1 0 0 − 1
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q3:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 𝜋12.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 − √ 2 2 √ 2 − √ 3 2 √ 3 − √ 2 2 1 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 √ 2 − √ 6 4 √ 6 − √ 2 4 √ 6 + √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 − √ 6 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ 1 − √ 2 2 √ 3 − √ 2 2 √ 3 − √ 2 2 1 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q4:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 2𝜋3.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 √ 3 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 − 1 2 − 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 − 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 3 2 √ 3 2 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q5:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 5𝜋12.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 − √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ − 1 − √ 2 2 √ 2 − √ 3 2 √ 3 − √ 2 2 − 1 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 − √ 2 4 − √ 2 − √ 6 4 − √ 2 − √ 6 4 √ 2 − √ 6 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 − √ 2 4 − √ 2 − √ 6 4 √ 6 + √ 2 4 √ 6 − √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − 1 − √ 2 2 √ 3 + √ 2 2 √ 3 − √ 2 2 − 1 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q6:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of −𝜋3.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 − √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 − 1 2 − 1 2 √ 3 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 3 2 1 2 − 1 2 √ 3 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 ℝ counterclockwise about the origin through an angle of 𝜋4.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 2 2 √ 2 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 − √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 − √ 2 2 √ 2 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q8:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 𝜋3.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 − √ 3 2 √ 3 2 − 1 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 ⎡ ⎢ ⎢ ⎢ ⎣ − √ 3 2 1 2 √ 3 2 1 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q9:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 𝜋12.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 √ 2 − √ 6 4 √ 6 − √ 2 4 √ 6 + √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 − √ 2 4 √ 2 − √ 6 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 − √ 6 + √ 2 4 √ 6 − √ 2 4 − √ 2 − √ 6 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 − √ 6 + √ 2 4 √ 6 + √ 2 4 √ 2 − √ 6 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 6 + √ 2 4 √ 2 − √ 6 4 √ 2 − √ 6 4 − √ 6 + √ 2 4 ⎤ ⎥ ⎥ ⎥ ⎦

Q10:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 5𝜋12.

  • 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 ⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦.

  • Aa rotation through an angle of 90∘
  • Ba rotation through an angle of 45∘
  • Ca rotation through an angle of −45∘
  • Da rotation through an angle of −90∘
  • Ea rotation through an angle of 135∘

Q12:

A rotation with center the origin sends the vector 34 to 43. Find the matrix representation of this rotation.

  • A ⎡ ⎢ ⎢ ⎣ 4 5 3 5 3 5 − 4 5 ⎤ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎣ 2 4 2 5 7 2 5 − 7 2 5 2 4 2 5 ⎤ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎣ 4 5 3 5 − 3 5 4 5 ⎤ ⎥ ⎥ ⎦
  • D 0 1 1 0
  • E ⎡ ⎢ ⎢ ⎣ 2 4 2 5 7 2 5 7 2 5 − 2 4 2 5 ⎤ ⎥ ⎥ ⎦

Q13:

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

  • A 𝜃 = 0 and 𝜃=𝜋
  • B 𝜃 = 0 and 𝜃=3𝜋2
  • C 𝜃 = 0 and 𝜃=𝜋3
  • D 𝜃 = 𝜋 2 and 𝜃=𝜋
  • E 𝜃 = 0 and 𝜃=𝜋2

Q14:

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

  • A 𝜃 𝜃 𝜃 𝜃 s i n c o s c o s s i n
  • B 𝜃 𝜃 − 𝜃 𝜃 c o s s i n s i n c o s
  • C − 𝜃 𝜃 − 𝜃 𝜃 c o s s i n s i n c o s
  • D 𝜃 − 𝜃 𝜃 𝜃 s i n c o s c o s s i n
  • E 𝜃 − 𝜃 𝜃 𝜃 c o s s i n s i n c o s

Q15:

Consider the linear transformation 𝐿∶ℝ→ℝ, where 𝐿 rotates each vector 90 degrees about the positive 𝑥-axis. Find the matrix representation of 𝐿 in the standard basis.

  • A 1 0 0 0 0 − 1 0 − 1 0
  • B 1 0 0 0 0 − 1 0 1 0
  • C 1 0 0 0 0 1 0 1 0
  • D 1 0 0 1 0 − 1 1 1 0

Q16:

A linear transformation of a plane sends the vector 10 to 𝑝𝑞. If the transformation is a rotation about the origin, where does it send 01?

  • A − 𝑞 𝑝
  • B 𝑝 𝑞
  • C − 𝑞 − 𝑝
  • D 𝑞 − 𝑝
  • E 𝑞 𝑝

Q17:

A linear transformation of a plane sends vector 10 to 𝑝𝑞. If the transformation is a rotation, where does it send 01?

  • A − 𝑞 − 𝑝
  • B 𝑞 𝑝
  • C 𝑝 𝑞
  • D − 𝑞 𝑝
  • E 𝑞 − 𝑝

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