# 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 which represents a rotation of by about an axis through the origin and in the direction of

What is ?

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

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The vector is perpendicular to . What can you say about the direction of the vector ?

• A will be perpendicular to but not necessarily to
• B will be parallel to
• C will be perpendicular to but not necessarily to
• D will be parallel to
• E will be perpendicular to both and

What can you say about the magnitude of ?

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

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What can you say about the vector ?

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

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Q2:

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.

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Q3:

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.

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