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Practice: Linear Transformation in Three Dimensions

In two-dimensional space you only need to specify the angle of a rotation. In three-dimensional space you need to give both an angle and a vector which represents the axis of rotation. Consider the matrix 𝐴 which represents a rotation of by 90 about an axis through the origin and in the direction of 𝑛=221.

What is 𝐴𝑛?

  • A010
  • B𝑛
  • C100
  • D001
  • E𝑛

Find the general form of the matrix which sends 𝑛 to the appropriate vector as determined in the previous part of the question.

  • A𝑎𝑏22𝑎2𝑏𝑐𝑑22𝑐2𝑑𝑒𝑓12𝑒2𝑓
  • B𝑎𝑏22𝑎2𝑏𝑐𝑑22𝑐2𝑑𝑒𝑓12𝑒2𝑓
  • C𝑎𝑏2𝑎2𝑏𝑐𝑑12𝑐2𝑑𝑒𝑓2𝑒2𝑓
  • D𝑎𝑏2𝑎2𝑏𝑐𝑑2𝑐2𝑑𝑒𝑓12𝑒2𝑓
  • E𝑎𝑏12𝑎2𝑏𝑐𝑑2𝑐2𝑑𝑒𝑓2𝑒2𝑓

The vector 𝑣=330 is perpendicular to 𝑛. What can you say about the direction of the vector 𝑤=𝐴𝑣?

  • A𝑤 will be perpendicular to both 𝑛 and 𝑣
  • B𝑤 will be parallel to 𝑛
  • C𝑤 will be perpendicular to 𝑣 but not necessarily to 𝑛
  • D𝑤 will be parallel to 𝑣
  • E𝑤 will be perpendicular to 𝑛 but not necessarily to 𝑣

What can you say about the magnitude of 𝑤=𝐴𝑣?

  • A|𝑤|=|𝑣|
  • B|𝑤|=1
  • C|𝑤|=3|𝑛|
  • D|𝑤|=3|𝑣|
  • E|𝑤|=|𝑛|

Which of the following vectors has the required properties to be 𝑤?

  • A3312
  • B114
  • C330
  • D114
  • E3312

What can you say about the vector 𝐴𝑤?

  • A𝐴𝑤=𝑤
  • B𝐴𝑤=𝑛
  • C𝐴𝑤=𝑣
  • D𝐴𝑤=𝑣
  • E𝐴𝑤=𝑛

Using the general form of the matrix from the second part of the question, and the values of 𝐴𝑣 and 𝐴𝑤, find the matrix 𝐴.

  • A231231104
  • B221330114
  • C19418744481
  • D154231231104
  • E418744481

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Incorrect Answer

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