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is characterized by the formula |
Question. If is the plane in that is the linear span of the vectors and is the rotation in space through about the axis through the origin that is perpendicular to , specify a basis of relative to which the matrix of is relatively simple.
Answer. There is slight ambiguity since it is not possible to distinguish between clockwise and counterclockwise.
is a basis of the plane
One computes the “dot product”:
So and are perpendicular.
One of the two possible rotations through will satisfy:
The “cross product” lies on the axis of rotation:
Take , a vector on the axis, as a third basis vector for .
With as selected pairs of bases, the matrix of is:
We have:
Note: The second matrix is the matrix of the linear map with respect to the basis pair .
The first matrix is not the matrix of a linear map but, rather, a matrix whose columns are the standard coordinates — coordinates with respect to the standard basis — of the members of the basis .
The matrix corresponding in a similar way to the standard basis is the identity matrix, and it would be more precise, instead of writing to use to relate the row of vectors to the row of vectors : is the matrix for change of basis between the basis and the standard basis .
For the standard matrix of one has or
Since is linear, and , one has and, therefore, combining the various formulas: yielding the following relation among ordinary matrices:
Because this particular matrix consists of mutually perpendicular columns, all of the same length, it is particularly easy to invert:
is the standard matrix of one of the two rotations through the angle about the line through the origin and the point .