Problem 37 - Precalculus: Mathematics for Calculus

Question

Let \(Z\), \(Z^{\prime},\) and \(R\) be the matrices $$\begin{array}{cc}Z=\left[\begin{array}{l}x \\\y\end{array}\right] & Z^{\prime}=\left[\begin{array}{l}X \\\Y\end{array}\right] \\\R=\left[\begin{array}{ll}\cos \phi & -\sin \phi \\\\\sin \phi & \cos \phi\end{array}\right] \end{array}$$ (a) Show that the Rotation of Axes Formulas can be written as $$Z=R Z^{\prime} \quad \text { and } \quad Z^{\prime}=R^{-1} Z$$ (b) Let \(R_{1}\) and \(R_{2}\) be matrices that represent rotations through the angles \(\phi_{1}\) and \(\phi_{2}\), respectively. Show that the product matrix \(R_{1} R_{2}\) represents a rotation through an angle \(\phi_{1}+\phi_{2} .\) IHint: Use the Addition Formulas for sine and cosine to simplify the entries of the matrix \(\left.R_{1} R_{2} .\right]\)

Step-by-Step Solution

Answer

Rotation matrices can express transformations and their compositions naturally. Using algebra and angle formulas, these changes can be confirmed.

Step 1: Matrix Multiplication for Rotation

The rotation of a vector in a 2D plane by an angle \(\phi\) can be expressed using a rotation matrix \(R\). The relation \(Z = R Z'\) shows that the coordinates \(Z\) are obtained by rotating the coordinates \(Z'\) by \(\phi\). Perform the matrix multiplication to show this:\[Z = \begin{pmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} X \ Y \end{pmatrix} = \begin{pmatrix} X \cos \phi - Y \sin \phi \ X \sin \phi + Y \cos \phi \end{pmatrix}\]This results in the rotated coordinates \(Z = (x, y)\).

Step 2: Find the Inverse Rotation Matrix

To reverse the rotation and find the original coordinates \(Z'\) from \(Z\), we use the inverse of the rotation matrix. The inverse of a rotation matrix is its transpose, \(R^{-1} = R^T\), because rotation matrices are orthogonal:\[R^{-1} = \begin{pmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{pmatrix}\]Now apply this inverse to express \(Z'\) in terms of \(Z\):\[Z' = R^{-1} Z = \begin{pmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} x \cos \phi + y \sin \phi \ -x \sin \phi + y \cos \phi \end{pmatrix}\]This recovers the original coordinates \(Z' = (X, Y)\).

Step 3: Define Rotation Matrices for Two Angles

Let \(R_1\) and \(R_2\) be rotation matrices for angles \(\phi_1\) and \(\phi_2\) respectively. They are:\[R_1 = \begin{pmatrix} \cos \phi_1 & -\sin \phi_1 \ \sin \phi_1 & \cos \phi_1 \end{pmatrix}, \quad R_2 = \begin{pmatrix} \cos \phi_2 & -\sin \phi_2 \ \sin \phi_2 & \cos \phi_2 \end{pmatrix}\]

Step 4: Express Product Matrix and Use Sine/Cosine Addition Formulas

The product matrix \(R_1 R_2\) corresponds to a rotation by angle \(\phi_1 + \phi_2\). Multiply \(R_1\) and \(R_2\):\[R_1 R_2 = \begin{pmatrix} \cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2 & -\cos \phi_1 \sin \phi_2 - \sin \phi_1 \cos \phi_2 \ \sin \phi_1 \cos \phi_2 + \cos \phi_1 \sin \phi_2 & -\sin \phi_1 \sin \phi_2 + \cos \phi_1 \cos \phi_2 \end{pmatrix}\]Using the sum formulas:\(\cos(\phi_1 + \phi_2) = \cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2, \ \sin(\phi_1 + \phi_2) = \sin \phi_1 \cos \phi_2 + \cos \phi_1 \sin \phi_2\), the matrix simplifies to\[R_1 R_2 = \begin{pmatrix} \cos(\phi_1 + \phi_2) & -\sin(\phi_1 + \phi_2) \ \sin(\phi_1 + \phi_2) & \cos(\phi_1 + \phi_2) \end{pmatrix}\]This demonstrates the rotation by \(\phi_1 + \phi_2\).

Key Concepts

Rotation of Axes

The rotation of axes involves transforming coordinates in a plane by rotating them through a certain angle. Imagine you have a set of points on the plane, and you want to "spin" these around a fixed origin. The rotation matrix is the mathematical tool used to facilitate this. It is expressed as:\[R = \begin{pmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{pmatrix}\]Here, \(\phi\) is the angle of rotation. To rotate a point \(Z' = (X, Y)\) to a new position \(Z = (x, y)\), you use the matrix multiplication \(Z = RZ'\). This applies a rotation of \(\phi\) degrees counterclockwise. The purpose of this transformation in various fields like engineering, graphics, and physics is to understand systems or shapes from different orientations.

Matrix Multiplication

Matrix multiplication is central to transforming coordinates, especially in rotations. When you multiply two matrices, you're essentially combining their transformations. For rotation, the matrix \(R\) provides a way to rotate the coordinates by performing multiplication with the coordinate vector:
\[Z = \begin{pmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} X \ Y \end{pmatrix} = \begin{pmatrix} X \cos \phi - Y \sin \phi \ X \sin \phi + Y \cos \phi \end{pmatrix}\]

• The first element of the result gives the new x-coordinate.
• The second element gives the new y-coordinate.

The beauty of matrix multiplication is that it allows for chaining transformations easily. If you have multiple transformations, you can multiply their matrices together to find the overall transformation effect.

Inverse of a Matrix

The inverse of a matrix is like an "un-do" button in transformations. For a rotation matrix, the inverse is simply its transpose. This is because rotation matrices are orthogonal, meaning their transpose is their inverse:
\[R^{-1} = R^T = \begin{pmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{pmatrix}\]
To "reverse" the rotation and get back to the original coordinates, you use this inverse:
\[Z' = R^{-1}Z\]

• This operation rotates the coordinates back by the angle \(\phi\), effectively reversing the initial transformation.

In many applications, being able to reverse operations like this is crucial, especially when maintaining data consistency or solving systems of equations.

Trigonometric Addition Formulas

The trigonometric addition formulas are key to understanding how combined rotations work. When you are dealing with two successive rotations, you can find the new composite angle using these formulas. Suppose you have angles \(\phi_1\) and \(\phi_2\):

• \(\cos(\phi_1 + \phi_2) = \cos \phi_1 \cos \phi_2 - \sin \phi_1 \sin \phi_2\)
• \(\sin(\phi_1 + \phi_2) = \sin \phi_1 \cos \phi_2 + \cos \phi_1 \sin \phi_2\)

These formulas allow you to derive the rotation matrix for combined angles:
\[R_1 R_2 = \begin{pmatrix}\cos(\phi_1 + \phi_2) & -\sin(\phi_1 + \phi_2) \ \sin(\phi_1 + \phi_2) & \cos(\phi_1 + \phi_2) \end{pmatrix}\]This demonstrates that performing multiple rotations is equivalent to a single rotation of the summed angles. It is vital in fields like navigation, robotics, and computer graphics where understanding compound movements is necessary.