Worksheet: Linear Transformation in Three Dimensions

In this worksheet, we will practice finding the matrix of single or composite linear transformations in three dimensions.

Q1:

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?

  • A001
  • B100
  • C𝑛
  • D𝑛
  • E010

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

  • A𝑎𝑏2𝑎2𝑏𝑐𝑑2𝑐2𝑑𝑒𝑓12𝑒2𝑓
  • B𝑎𝑏12𝑎2𝑏𝑐𝑑2𝑐2𝑑𝑒𝑓2𝑒2𝑓
  • C𝑎𝑏2𝑎2𝑏𝑐𝑑12𝑐2𝑑𝑒𝑓2𝑒2𝑓
  • D𝑎𝑏22𝑎2𝑏𝑐𝑑22𝑐2𝑑𝑒𝑓12𝑒2𝑓
  • E𝑎𝑏22𝑎2𝑏𝑐𝑑22𝑐2𝑑𝑒𝑓12𝑒2𝑓

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 𝑛 but not necessarily to 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 both 𝑛 and 𝑣

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

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

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

  • A3312
  • B114
  • C3312
  • D330
  • E114

What can you say about the vector 𝐴𝑤?

  • A𝐴𝑤=𝑤
  • B𝐴𝑤=𝑣
  • C𝐴𝑤=𝑛
  • 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.

  • A418744481
  • B221330114
  • C231231104
  • D19418744481
  • E154231231104

Q2:

Consider the linear transformation 𝐿, which rotates each vector 90 degrees counterclockwise about the positive 𝑥-axis then 45 degrees counterclockwise about the positive 𝑦-axis. Find the matrix representation of 𝐿 in the standard basis.

  • A1212000112120
  • B1212000112120
  • C121200011120
  • D1212010112120

Q3:

Consider the linear transformation 𝐿, which rotates each vector 45 degrees counterclockwise about the positive 𝑥-axis then 90 degrees counterclockwise about the positive 𝑦-axis. Find 𝐿121 in the standard basis.

  • A12+21221
  • B12+221
  • C12+2112+2
  • D121

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