Worksheet: Linear Transformations in Planes: Rotation

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 โ„๏Šจ through an angle of ๐œ‹3.

  • AโŽกโŽขโŽขโŽขโŽฃ12โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • C๏”โˆ’100โˆ’1๏ 
  • DโŽกโŽขโŽขโŽขโŽฃโˆš32โˆ’1212โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆš321212โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ

Q2:

Find the matrix for the linear transformation that rotates every vector in โ„๏Šจ through an angle of ๐œ‹4.

  • AโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • B๏”โˆ’100โˆ’1๏ 
  • CโŽกโŽขโŽขโŽขโŽฃโˆš22โˆš22โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃ12โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆš22โˆ’โˆš22โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ

Q3:

Find the matrix for the linear transformation which rotates every vector in โ„๏Šจ through an angle of ๐œ‹12.

  • AโŽกโŽขโŽขโŽขโŽฃ1โˆ’โˆš22โˆš2โˆ’โˆš32โˆš3โˆ’โˆš221โˆ’โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆš2โˆ’โˆš64โˆš6โˆ’โˆš24โˆš6+โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš2โˆ’โˆš64โˆš6+โˆš24โˆš6+โˆš24โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃ1โˆ’โˆš22โˆš3โˆ’โˆš22โˆš3โˆ’โˆš221โˆ’โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ

Q4:

Find the matrix for the linear transformation which rotates every vector in โ„๏Šจ through an angle of 2๐œ‹3.

  • AโŽกโŽขโŽขโŽขโŽฃโˆ’12โˆš32โˆš32โˆ’12โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš32โˆ’12โˆ’12โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš3212โˆ’12โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆ’12โˆ’โˆš32โˆš32โˆ’12โŽคโŽฅโŽฅโŽฅโŽฆ

Q5:

Find the matrix for the linear transformation which rotates every vector in โ„๏Šจ through an angle of 5๐œ‹12.

  • AโŽกโŽขโŽขโŽขโŽฃโˆš6โˆ’โˆš24โˆš6+โˆš24โˆš6+โˆš24โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃโˆ’1โˆ’โˆš22โˆš2โˆ’โˆš32โˆš3โˆ’โˆš22โˆ’1โˆ’โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆš6โˆ’โˆš24โˆ’โˆš2โˆ’โˆš64โˆ’โˆš2โˆ’โˆš64โˆš2โˆ’โˆš64โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš6โˆ’โˆš24โˆ’โˆš2โˆ’โˆš64โˆš6+โˆš24โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆ’1โˆ’โˆš22โˆš3+โˆš22โˆš3โˆ’โˆš22โˆ’1โˆ’โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ

Q6:

Find the matrix for the linear transformation which rotates every vector in โ„๏Šจ through an angle of โˆ’๐œ‹3.

  • AโŽกโŽขโŽขโŽขโŽฃ12โˆš32โˆ’โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆ’โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆš32โˆ’12โˆ’12โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš3212โˆ’12โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ

Q7:

Find, with respect to the standard basis, the matrix which rotates every vector in โ„๏Šจ counterclockwise about the origin through an angle of ๐œ‹4.

  • AโŽกโŽขโŽขโŽขโŽฃโˆš22โˆ’โˆš22โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš22โˆš2212โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆš22โˆš22โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš22โˆ’โˆš22โˆ’โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆ’12โˆ’โˆš22โˆš2212โŽคโŽฅโŽฅโŽฅโŽฆ

Q8:

Find, with respect to the standard basis, the matrix which rotates every vector in โ„๏Šจ counterclockwise about the origin through an angle of ๐œ‹3.

  • AโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš32โˆ’12โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃ12โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš3212โˆ’12โˆš32โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃ12โˆ’โˆš32โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš3212โˆš3212โŽคโŽฅโŽฅโŽฅโŽฆ

Q9:

Find, with respect to the standard basis, the matrix which rotates every vector in โ„๏Šจ counterclockwise about the origin through an angle of ๐œ‹12.

  • AโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆš2โˆ’โˆš64โˆš6โˆ’โˆš24โˆš6+โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆš6+โˆš24โˆš6โˆ’โˆš24โˆš2โˆ’โˆš64โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆ’โˆš6+โˆš24โˆš6โˆ’โˆš24โˆ’โˆš2โˆ’โˆš64โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆ’โˆš6+โˆš24โˆš6+โˆš24โˆš2โˆ’โˆš64โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆš2โˆ’โˆš64โˆš2โˆ’โˆš64โˆ’โˆš6+โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ

Q10:

Find, with respect to the standard basis, the matrix which rotates every vector in โ„๏Šจ counterclockwise about the origin through an angle of 5๐œ‹12.

  • AโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš6+โˆš24โˆ’โˆš6+โˆš24โˆ’โˆš6+โˆš24โˆ’โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽขโŽฃโˆš6โˆ’โˆš24โˆ’โˆš6+โˆš24โˆš6+โˆš24โˆ’โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽขโŽฃโˆ’โˆš6โˆ’โˆš24โˆš6+โˆš24โˆš6+โˆš24โˆ’โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽขโŽฃโˆš6+โˆš24โˆ’โˆš6+โˆš24โˆ’โˆš6+โˆš24โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽขโŽฃโˆš6โˆ’โˆš24โˆ’โˆš6+โˆš24โˆš6+โˆš24โˆš6โˆ’โˆš24โŽคโŽฅโŽฅโŽฅโŽฆ

Q11:

Describe the geometric effect of the transformation produced by the matrix โŽกโŽขโŽขโŽขโŽฃโˆš22โˆ’โˆš22โˆš22โˆš22โŽคโŽฅโŽฅโŽฅโŽฆ.

  • Aa rotation through an angle of 90โˆ˜
  • Ba rotation through an angle of 45โˆ˜
  • Ca rotation through an angle of โˆ’45โˆ˜
  • Da rotation through an angle of โˆ’90โˆ˜
  • Ea rotation through an angle of 135โˆ˜

Q12:

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

  • AโŽกโŽขโŽขโŽฃ453535โˆ’45โŽคโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽฃ2425725โˆ’7252425โŽคโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽฃ4535โˆ’3545โŽคโŽฅโŽฅโŽฆ
  • D๏”0110๏ 
  • EโŽกโŽขโŽขโŽฃ2425725725โˆ’2425โŽคโŽฅโŽฅโŽฆ

Q13:

Let ๐ด be the matrix of the transformation which rotates all vectors in โ„๏Šจ through an angle of ๐œƒ, where 0<๐œƒโ‰ค2๐œ‹. For which values of ๐œƒ does ๐ด have a real eigenvalue?

  • A๐œƒ=0 and ๐œƒ=๐œ‹
  • B๐œƒ=0 and ๐œƒ=3๐œ‹2
  • C๐œƒ=0 and ๐œƒ=๐œ‹3
  • D๐œƒ=๐œ‹2 and ๐œƒ=๐œ‹
  • E๐œƒ=0 and ๐œƒ=๐œ‹2

Q14:

Consider the linear map ๐‘‡โˆถโ„โ†’โ„๏Šจ๏Šจ, which acts by rotating a vector ๐œƒโˆ˜ counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

  • A๏”๐œƒ๐œƒ๐œƒ๐œƒ๏ sincoscossin
  • B๏”๐œƒ๐œƒโˆ’๐œƒ๐œƒ๏ cossinsincos
  • C๏”โˆ’๐œƒ๐œƒโˆ’๐œƒ๐œƒ๏ cossinsincos
  • D๏”๐œƒโˆ’๐œƒ๐œƒ๐œƒ๏ sincoscossin
  • E๏”๐œƒโˆ’๐œƒ๐œƒ๐œƒ๏ cossinsincos

Q15:

Consider the linear transformation ๐ฟโˆถโ„โ†’โ„๏Šฉ๏Šฉ, where ๐ฟ rotates each vector 90 degrees about the positive ๐‘ฅ-axis. Find the matrix representation of ๐ฟ in the standard basis.

  • A๏˜10000โˆ’10โˆ’10๏ค
  • B๏˜10000โˆ’1010๏ค
  • C๏˜100001010๏ค
  • D๏˜10010โˆ’1110๏ค

Q16:

A linear transformation of a plane sends the vector ๏”10๏  to ๏”๐‘๐‘ž๏ . If the transformation is a rotation about the origin, where does it send ๏”01๏ ?

  • A๏”โˆ’๐‘ž๐‘๏ 
  • B๏”๐‘๐‘ž๏ 
  • C๏”โˆ’๐‘žโˆ’๐‘๏ 
  • D๏”๐‘žโˆ’๐‘๏ 
  • E๏”๐‘ž๐‘๏ 

Q17:

A linear transformation of a plane sends vector ๏”10๏  to ๏”๐‘๐‘ž๏ . If the transformation is a rotation, where does it send ๏”01๏ ?

  • A๏”โˆ’๐‘žโˆ’๐‘๏ 
  • B๏”๐‘ž๐‘๏ 
  • C๏”๐‘๐‘ž๏ 
  • D๏”โˆ’๐‘ž๐‘๏ 
  • E๏”๐‘žโˆ’๐‘๏ 

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