The matrix represents a linear transformation which sends the vector to . What can you say about the matrix ?
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 ?
Find the general form of the matrix which sends to the appropriate vector as determined in the previous part of the question.
The vector is perpendicular to . What can you say about the direction of the vector ?
What can you say about the magnitude of ?
Which of the following vectors has the required properties to be ?
What can you say about the vector ?
Using the general form of the matrix from the second part of the question, and the values of and , find the matrix .
Consider the linear transformations for which , the image of , and , the image of , are unit vectors. Let be a linear transformation of this kind which has the additional property that the area of the parallelogram with vertices , , , and is as big as possible. What are the possible values of the measure of the angle between and for the transformation ?