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Lesson: Rotation Matrices

Worksheet • 16 Questions

Q1:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ο€Ό βˆ’ 1 0 0 βˆ’ 1 
  • C βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ √ 3 2 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ √ 3 2 βˆ’ 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q2:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ √ 2 2 βˆ’ √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ο€Ό βˆ’ 1 0 0 βˆ’ 1 
  • C βŽ› ⎜ ⎜ ⎜ ⎝ √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q3:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 + √ 2 4 √ 2 βˆ’ √ 6 4 √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎝ √ 2 βˆ’ √ 6 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ 1 βˆ’ √ 2 2 √ 3 βˆ’ √ 2 2 √ 3 βˆ’ √ 2 2 1 βˆ’ √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ 1 βˆ’ √ 2 2 √ 2 βˆ’ √ 3 2 √ 3 βˆ’ √ 2 2 1 βˆ’ √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q4:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ 1 2 βˆ’ √ 3 2 √ 3 2 βˆ’ 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ 1 2 √ 3 2 √ 3 2 βˆ’ 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ √ 3 2 1 2 βˆ’ 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ √ 3 2 βˆ’ 1 2 βˆ’ 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q5:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 2 βˆ’ √ 6 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ 1 βˆ’ √ 2 2 √ 3 + √ 2 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ 1 βˆ’ √ 2 2 √ 2 βˆ’ √ 3 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q6:

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

  • A βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 βˆ’ √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎝ 1 2 βˆ’ √ 3 2 βˆ’ √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ √ 3 2 βˆ’ 1 2 βˆ’ 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ √ 3 2 1 2 βˆ’ 1 2 √ 3 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œ‹ 1 2 .

  • A βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 βˆ’ √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎝ βˆ’ √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎝ √ 6 βˆ’ √ 2 4 βˆ’ √ 6 + √ 2 4 √ 6 + √ 2 4 βˆ’ √ 6 βˆ’ √ 2 4 ⎞ ⎟ ⎟ ⎟ ⎠

Q11:

Describe the geometric effect of the transformation produced by the matrix βŽ› ⎜ ⎜ ⎜ ⎝ √ 2 2 βˆ’ √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠ .

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

Q12:

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

  • A βŽ› ⎜ ⎜ ⎝ 2 4 2 5 7 2 5 βˆ’ 7 2 5 2 4 2 5 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ 4 5 3 5 3 5 βˆ’ 4 5 ⎞ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎝ 4 5 3 5 βˆ’ 3 5 4 5 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎝ 2 4 2 5 7 2 5 7 2 5 βˆ’ 2 4 2 5 ⎞ ⎟ ⎟ ⎠
  • E ο€Ό 0 1 1 0 

Q13:

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

Q14:

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

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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