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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, with respect to the standard basis, the matrix which rotates every vector in ℝ 2 counterclockwise about the origin through an angle of πœ‹ 3 .

  • A ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 3 2 1 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 βˆ’ √ 3 2 √ 3 2 βˆ’ 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 3 2 1 2 βˆ’ 1 2 √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q2:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ  counterclockwise about the origin through an angle of πœ‹ 4 .

  • A ⎑ ⎒ ⎒ ⎒ ⎣ √ 2 2 βˆ’ √ 2 2 βˆ’ √ 2 2 √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 βˆ’ √ 2 2 √ 2 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ √ 2 2 βˆ’ √ 2 2 √ 2 2 √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 2 βˆ’ √ 2 2 √ 2 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q3:

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œ‹ 1 2 .

  • A ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q4:

Describe the geometric effect of the transformation produced by the matrix ⎑ ⎒ ⎒ ⎒ ⎣ √ 2 2 βˆ’ √ 2 2 √ 2 2 √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦ .

  • Aa rotation through an angle of βˆ’ 9 0 ∘
  • Ba rotation through an angle of βˆ’ 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 ∘

Q5:

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

  • A
  • B
  • C
  • D
  • E

Q6:

Consider the linear map 𝑇 ∢ ℝ β†’ ℝ 2 2 , which acts by rotating a vector πœƒ ∘ counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

  • A
  • B
  • C
  • D
  • E

Q7:

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

  • A
  • B
  • C
  • D
  • E

Q8:

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

  • A
  • B
  • C
  • D
  • E

Q9:

A rotation with center 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 βˆ’ 3 5 4 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 7 2 5 βˆ’ 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 βˆ’ 7 2 5 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 4 5 3 5 3 5 βˆ’ 4 5 ⎀ βŽ₯ βŽ₯ ⎦

Q10:

Consider the linear transformation 𝐿 β€² ∢ ℝ β†’ ℝ 3 3 , which rotates each vector 90 degrees counterclockwise about the positive π‘₯ -axis then 45 degrees counterclockwise about the positive 𝑦 -axis. Find the matrix representation of 𝐿 β€² in the standard basis.

  • A ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 βˆ’ 1 √ 2 0 0 0 βˆ’ 1 βˆ’ 1 √ 2 1 √ 2 0 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 βˆ’ 1 √ 2 0 0 0 βˆ’ 1 βˆ’ 1 1 √ 2 0 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 2 0 1 0 βˆ’ 1 1 √ 2 βˆ’ 1 √ 2 0 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ 1 √ 2 1 √ 2 0 0 0 βˆ’ 1 βˆ’ 1 √ 2 1 √ 2 0 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦

Q11:

Consider the linear transformation 𝐿 β€² ∢ ℝ β†’ ℝ 3 3 , which rotates each vector 45 degrees counterclockwise about the positive π‘₯ -axis then 90 degrees counterclockwise about the positive 𝑦 -axis. Find in the standard basis.

  • A ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 + √ 2 2 βˆ’ 1 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B  1 2 βˆ’ 1 
  • C ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 + √ 2 βˆ’ 1 √ 2 βˆ’ √ 2 1 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ 1 √ 2 + √ 2 1 βˆ’ 1 √ 2 + √ 2 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦

Q12:

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

  • A
  • B
  • C
  • D
  • E

Q13:

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

  • A ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 βˆ’ √ 2 2 √ 2 βˆ’ √ 3 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 βˆ’ √ 2 2 √ 3 + √ 2 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 2 βˆ’ √ 6 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q14:

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

  • A πœƒ = 0 and πœƒ = 3 πœ‹ 2
  • B πœƒ = 0 and πœƒ = πœ‹ 2
  • C πœƒ = 0 and πœƒ = πœ‹ 3
  • D πœƒ = 0 and πœƒ = πœ‹
  • E πœƒ = πœ‹ 2 and πœƒ = πœ‹

Q15:

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

  • A
  • B
  • C
  • D
  • E

Q16:

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

  • A ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 √ 2 βˆ’ √ 6 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 + √ 2 4 √ 2 βˆ’ √ 6 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 √ 2 βˆ’ √ 6 4 √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 + √ 2 4 √ 2 βˆ’ √ 6 4 √ 2 βˆ’ √ 6 4 βˆ’ √ 6 + √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

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