Linear Algebra and Its Applications, Exercise 2.6.21

Exercise 2.6.21. Consider the transformation from \mathbb{R}^3 to \mathbb{R}^3 that takes (x_1, x_2, x_3) into (x_2, x_3, x_1). What is the axis of rotation for the transformation? What is the angle of rotation?

Answer: We can approach this problem in at least two ways. The first way is to look at the effect that the transformation has on the unit vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) along the x, y, and z axes respectively.

To begin with, the transformation will send (0, 1, 0) to (1, 0, 0), in essence taking all points on the y-axis to the x-axis. One rotation that does this is the rotation through 90 degrees with the z-axis as the axis of rotation; however this rotation would send (0, 0, 1) to (0, 0, 1) (in other words, leave it unchanged), which is incorrect. Another possible rotation is the rotation through 180 degrees with the axis of rotation being the 45-degree line y = x in the xy plane; However this rotation would send (0, 0, 1) to (0, 0, -1), which is also incorrect. We conclude that the axis of rotation must be somewhere between the xy plane and the z-axis.

Similarly, the transformation will send (0, 0, 1) to (0, 1, 0), taking points on the z-axis to the y-axis. Two possible rotations that do this are the rotation through 90 degrees with the x-axis as the axis of rotation and the rotation through 180 degrees with the axis of rotation being the 45-degree line z = y in the yz plane; However these rotations do not correctly transform the vector (0, 0, 1) and so we conclude that the axis of rotation must be somewhere between the yz plane and the x-axis.

A similar argument based on the transformation sending (1, 0, 0) to (0, 0, 1) leads us to conclude that the axis of rotation must be somewhere between the xz plane and the y-axis.

So the axis of rotation must lie off the coordinate axes and the coordinate planes, and since the transformation is symmetric with respect to the coordinates we conclude that the axis of rotation is at an equal distance (in terms of degrees) from each of the coordinate axes and planes. The obvious candidate for the axis of rotation is then the line x = y = z that passes through the origin and the point (1, 1, 1), and is at an angle of 45 degrees from each of the axes and coordinate planes.

The second and simpler way to determine the axis of rotation is to recall that points on the axis of rotation remain unchanged by a rotation transformation. The transformation in question simply shifts each value to the left one position, so if the three values in a vector are all equal then the vector would remain unchanged by the transformation. We therefore conclude that the axis of rotation is the line for which x = y= z in agreement with the argument above.

Note that applying the transformation once sends (x_1, x_2, x_3) into (x_2, x_3, x_1), applying it again to the resulting vector sends (x_2, x_3, x_1) into (x_3, x_1, x_2), and applying it a third time sends (x_3, x_1, x_2) into (x_1, x_2, x_3), the original vector. Applying the transformation three times thus corresponds to a rotation of 360 degrees, so applying the transformation once corresponds to a rotation of 120 degrees.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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