Understanding Rotation in the YZ-Plane About the X-Axis and Trigonometric Functions

Introduction to X-Axis Rotation

When we consider the rotation of a point about the X-axis in the three-dimensional space ( mathbb{R}^3 ), we often use a transformation matrix. For a simple case, where the vector (01) (which lies along the Y-axis) is rotated counterclockwise by an angle (a), we observe some interesting behavior.

The rotation matrix for a counterclockwise rotation about the X-axis is given by:

[begin{pmatrix} 1 0 0 0 cos a -sin a 0 sin a cos a end{pmatrix}]

Effect of Rotation on the Unit Vector 01

Let's consider the unit vector (01). When this vector is rotated about the X-axis by an angle (a), its (x)-component remains the same because the rotation axis is the X-axis. However, the (y)- and (z)-components will change.

The vector (01) is transformed to:

[begin{pmatrix} 0 0 1 end{pmatrix} rightarrow begin{pmatrix} 0 -sin a cos a end{pmatrix}]

This means the vector's (y)-component becomes (-sin a), and the (z)-component becomes (cos a).

Reconciliation with Trigonometric Functions

The fact that the (y)-component of the vector is (-sin a) can initially seem counterintuitive because it lies in the third quadrant where the sine function is negative. However, this is because we are considering the rotation into the YZ-plane, not just the standard Y-axis projection.

It is important to note that (sin a) is a sine function value, which is always positive for angles in the first and second quadrants. When we rotate the vector (01) by an angle (a) into the YZ-plane, the boundaries of the rotation into the proper quadrant must be correctly understood. For instance, if we rotate the vector (01) by (90) degrees, it moves to the point (0 -1 0), which is indeed in the third quadrant.

Sketching the Rotation

To visualize the rotation, we can sketch the vector (01) in the YZ-plane and observe how it changes as it rotates. The rotation matrix transforms the vector as follows:

[begin{pmatrix} 0 0 1 end{pmatrix} xrightarrow{text{Rotation by } a text{ degrees}} begin{pmatrix} 0 -sin a cos a end{pmatrix}]

Sketching the vectors, you will notice that the (y)-coordinate changes from (0) to (-sin a), and the (z)-coordinate changes from (1) to (cos a).

The Importance of Quadrants in Rotation

Trigonometric functions like sine and cosine are periodic and have specific signs depending on the quadrant in which the angle lies. Hence, the transformation of the vector (01) to (0 -sin a cos a) aligns with the properties of trigonometric functions in the appropriate quadrants. This is not a contradiction but rather a reflection of the correct quadrant and the correct values of the trigonometric functions.

For example, in the first quadrant (0 to 90 degrees), (sin a) is positive, and in the second quadrant (90 to 180 degrees), (sin a) is still positive. Thus, when we rotate by 90 degrees, the value (-sin a) in the Y-coordinate ensures the correct position in the third quadrant.

Conclusion

In summary, the rotation of the vector (01) about the X-axis and the resulting values of the transformed vector's components are consistent with the trigonometric functions and the specific quadrants involved. Understanding these concepts helps in correctly interpreting and visualizing vector transformations in three-dimensional space.