Problem 37 - Precalculus Mathematics for Calculus

Question

Let \(Z, Z^{\prime},\) and \(R\) be the matrices $$\begin{array}{cc} Z=\left[\begin{array}{l}x \\\y\end{array}\right] &Z^{\prime}=\left[\begin{array}{l}X \\\Y\end{array}\right] \\\R=\left[\begin{array}{ll}\cos \phi & -\sin \phi \\\\\sin \phi & \cos \phi\end{array}\right]\end{array}$$ Show that the Rotation of Axes Formulas can be written as $$Z=R Z^{\prime} \quad \text { and } \quad Z^{\prime}=R^{-1} Z$$

Step-by-Step Solution

Answer

\( Z = R Z' \) and \( Z' = R^{-1} Z \) show rotation transformations.

Step 1: Understand the Matrices

First, we identify each matrix and its components. Here, \( Z \) and \( Z' \) are vectors, represented as column matrices with two elements each. \( R \) is a 2x2 rotation matrix with angle \( \phi \), where \( R = \begin{bmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{bmatrix} \).

Step 2: Express Z using R and Z'

To show that \( Z = R Z' \), perform matrix multiplication between \( R \) and \( Z' \). Compute \( R Z' = \begin{bmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} X \ Y \end{bmatrix} \). The resulting matrix is \( Z = \begin{bmatrix} X \cos \phi - Y \sin \phi \ X \sin \phi + Y \cos \phi \end{bmatrix} \), which shows that \( Z = R Z' \).

Step 3: Find the inverse of R

The inverse of a rotation matrix \( R \) is its transpose, given by \( R^{-1} = \begin{bmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{bmatrix} \). This is because rotation matrices are orthogonal.

Step 4: Express Z' using R^{-1} and Z

To show that \( Z' = R^{-1} Z \), perform matrix multiplication between \( R^{-1} \) and \( Z \). Compute \( R^{-1} Z = \begin{bmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \). The resulting matrix is \( Z' = \begin{bmatrix} x \cos \phi + y \sin \phi \ -x \sin \phi + y \cos \phi \end{bmatrix} \), confirming \( Z' = R^{-1} Z \).

Step 5: Conclusion

The matrix equations \( Z = R Z' \) and \( Z' = R^{-1} Z \) confirm the rotation of axes formulas for transforming between coordinates \( (x, y) \) and the rotated coordinates \( (X, Y) \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation used in mathematics, especially in linear algebra. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Here, we're dealing with the multiplication of a 2x2 matrix by a 2x1 matrix, which results in another 2x1 matrix. This operation is vital when handling transformations such as rotations or scaling in geometry. The formula for multiplying matrices is given by the sum of the products of rows and columns.

For example, when we multiply the rotation matrix \( R = \begin{bmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{bmatrix} \) by the vector \( Z' = \begin{bmatrix} X \ Y \end{bmatrix} \), the result is:

• First row: \( \cos \phi \cdot X + (-\sin \phi) \cdot Y \)
• Second row: \( \sin \phi \cdot X + \cos \phi \cdot Y \)

Thus, matrix multiplication allows us to express the rotation of coordinates elegantly, showing a direct relationship between the initial vector and the rotated vector.

Inverse Matrix

The inverse matrix is a key concept in linear algebra that reverses the effect of a matrix transformation. For a square matrix \( A \), its inverse is denoted as \( A^{-1} \) and satisfies the condition \( A A^{-1} = A^{-1} A = I \), where \( I \) is the identity matrix. The identity matrix acts like the number 1 in matrix algebra, leaving other matrices unchanged when multiplied by them.

For the rotation matrix \( R \), the inverse \( R^{-1} \) is especially handy because the rotation matrix is orthogonal. This means its transpose is also its inverse:

• \( R^{-1} = \begin{bmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{bmatrix} \)

Using the inverse matrix, we can change back from the rotated coordinates \( (X, Y) \) to the original coordinates \( (x, y) \). This mirrors the essence of performing and undoing transformations, allowing us to switch seamlessly between different coordinate perspectives.

Coordinate Transformation

Coordinate transformation is crucial for translating geometric problems between different frames of reference. It adapts coordinates from one system to another by applying transformations such as rotations, translations, or scaling.

The matrix \( R \), a rotation matrix, helps achieve this by rotating coordinates counterclockwise through an angle \( \phi \). When you multiply \( R \) by \( Z' \), it transforms the coordinates \( (X, Y) \) into the rotated frame, resulting in \( (x, y) \). Conversely, multiplying by the inverse \( R^{-1} \) returns the system back to its original orientation.

This transformation process is significant in various fields such as computer graphics, navigation, and robotics, where understanding different perspectives and orientations can be essential. It enables the conversion of data from sensors to a coordinate system that an application can use effectively, making mathematics a powerful tool in real-world application.