This post describes how Maple was used to investigate the Givens rotation matrix, and to answer a simple question about its behavior. The "Givens" part is the medium, but the message is that it really is better to teach, learn, and do mathematics with a tool like Maple.

**The question:** If Givens rotations are used to take the vector **Y** = <5, -2, 1> to **Y**_{2} = , about what axis and through what angle will a single rotation accomplish the same thing?

The Givens matrix *G*_{21 }takes **Y** to the vector **Y**_{1} =, and the Givens matrix *G*_{31 }takes **Y**_{1 }to **Y**_{2}. Graphing the vectors **Y**, **Y**_{1}, and **Y**_{2} reveals that **Y**_{1 }lies in the xz-plane and that **Y**_{2} is parallel to the x-axis. (These geometrical observations should have been obvious, but the typical usage of the Givens technique to "zero-out" elements in a vector or matrix obscured this, at least for me.)

The matrix *G = G*_{31} G_{21 }rotates **Y** directly to **Y**_{2}; is the axis of rotation the vector **W = Y x Y**_{2}, and is the angle of rotation the angle between **Y** and **Y**_{2}? To test these hypotheses, I used the RotationMatrix command in the Student LinearAlgebra package to build the corresponding rotation matrix *R*. But *R* did not agree with *G*. I had either the axis or the angle (actually both) incorrect.

The individual Givens rotation matrices are orthogonal, so *G*, their product is also orthogonal. It will have 1 as its single real eigenvalue, and the corresponding eigenvector **V** is actually the direction of the axis of the rotation. The vector **W** is a multiple of <0, 1, 2> but **V** = <a, b, 1>, where . Clearly, **W** **V**.

The rotation matrix that rotates about the axis **V** through the angle isn't the matrix *G* either. The correct angle of rotation about **V** turns out to be

the angle between the projections of **Y** and **Y**_{2} onto the plane orthogonal to **V**. That came as a great surprise, one that required a significant adjustment of my intuition about spatial rotations. So again, the message is that teaching, learning, and doing mathematics is so much more effective and efficient when done with a tool like Maple.

A discussion of the Givens rotation, and a summary of the actual computations described above are available in the attached worksheet, What Gives with Givens.mw.