In this lesson, we will learn how to find 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 ℝ by 90∘ about an axis through the origin and in the direction of 𝑛=221.

What is 𝐴𝑛?

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

The vector 𝑣=3−30 is perpendicular to 𝑛. What can you say about the direction of the vector 𝑤=𝐴𝑣?

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

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

What can you say about the vector 𝐴𝑤?

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.

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