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Magazine Chapter 11: Problem 35
Let \(Z, Z^{\prime},\) and \(R\) be the matrices $$Z=\left[\begin{array}{l}{x} \\ {y}\end{array}\right] \quad Z^{\prime}=\left[\begin{array}{l}{X} \\ {Y}\end{array}\right]$$ $$R=\left[\begin{array}{cc}{\cos \phi} & {-\sin \phi} \\ {\sin \phi} & {\cos \phi}\end{array}\right]$$ Show that the Rotation of Axes Formulas can be written as $$Z=R Z^{\prime} \quad \text { and } \quad Z^{\prime}=R^{-1} Z$$
Short Answer
Expert verified
For rotation formulas: \(Z = R Z'\) and \(Z' = R^{-1} Z\).
Step by step solution
01
Understand the Rotation Matrix
The matrix \(R\) is a rotation matrix that rotates a point in the plane counterclockwise by an angle \( \phi \). The elements of \(R\) are trigonometric functions: \( \cos \phi \) and \( \sin \phi \).
02
Set up the Matrix Equation for Rotation
According to the problem, we are to show that \( Z = R Z' \). This implies that if you multiply the rotation matrix \(R\) by the original coordinates \(Z'\), you will obtain the new coordinates \(Z\).
03
Matrix Multiplication
Carry out the matrix multiplication for \(Z = R Z'\):\[\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} X \ Y \end{bmatrix}\]Multiply the matrices:- \(x = \cos \phi \cdot X - \sin \phi \cdot Y\)- \(y = \sin \phi \cdot X + \cos \phi \cdot Y\)
04
Show the Inverse Rotation
Now, to show that \( Z' = R^{-1} Z \), we first find \(R^{-1}\). The inverse of a rotation matrix is its transpose, because rotation matrices are orthogonal. Thus,\[R^{-1} = R^T = \begin{bmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{bmatrix}\]Hence, the equation \(Z' = R^{-1} Z\) becomes:\[\begin{bmatrix} X \ Y \end{bmatrix} = \begin{bmatrix} \cos \phi & \sin \phi \ -\sin \phi & \cos \phi \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}\]Perform the matrix multiplication:- \(X = \cos \phi \cdot x + \sin \phi \cdot y\)- \(Y = -\sin \phi \cdot x + \cos \phi \cdot y\)
05
Confirm Identity in Both Directions
Check that substituting \(X, Y\) into the equations for \(x, y\) gives identities, and verify the equations satisfy the rotation transformations. This confirms that \(Z = R Z'\) and \(Z' = R^{-1} Z\) represent a rotation of axes.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Multiplication
Matrix multiplication is a fundamental concept in linear algebra. It's crucial for transforming coordinates and solving various problems in mathematics and engineering. When you multiply two matrices, you combine rows and columns in a specific way.
Here's how it works:
Here's how it works:
- Take the rows from the first matrix and the columns from the second matrix.
- Multiply each pair of corresponding entries and sum them up.
- This sum becomes an element of the resulting matrix.
Inverse Matrix
An inverse matrix is like the reverse operation of matrix multiplication. If you have a matrix, its inverse, when multiplied by the original matrix, will result in an identity matrix.
This is much like the concept of division in arithmetic:
This is much like the concept of division in arithmetic:
- Not all matrices have inverses, only square matrices with non-zero determinants.
- The inverse of a rotation matrix is particularly interesting because it is also a rotation matrix.
- For a rotation matrix, its inverse is simply its transpose. This simplifies calculations and ensures that the original coordinates can be retrieved after a transformation.
Trigonometric Functions
Trigonometric functions like sine (\( \sin \)) and cosine (\( \cos \)) are used to relate angles to ratios of sides in right triangles. These functions are essential in defining the elements of rotation matrices.
Here's why they are important:
Here's why they are important:
- The functions help determine the proper scaling and direction needed for rotation.
- In a rotation matrix, \( \cos(\phi) \) controls the horizontal movement, while \( \sin(\phi) \) controls the vertical movement.
- These functions use angles to dictate the extent of rotation and are fundamental in the derivation and application of rotation formulas.
Orthogonal Matrix
An orthogonal matrix is one where its rows and columns are orthogonal unit vectors. This means the matrix is particularly well-behaved when it comes to inverses and computations.
Key characteristics include:
Key characteristics include:
- Orthogonal matrices have their inverse equal to their transpose, making calculations simpler.
- They preserve the length of vectors, which means transformations do not distort the size of shapes.
- In the context of rotation, orthogonal matrices elegantly represent rotations because they maintain the structure of the space.
Rotation of Axes
The rotation of axes is a transformation that changes the orientation of the coordinate system while preserving the relative geometry of objects within that system. When axes are rotated, coordinates must be transformed accordingly.
Important points about rotation of axes include:
Important points about rotation of axes include:
- Rotation is achieved through the use of rotation matrices, which are defined by angle \(\phi\).
- The operation can be reversed using the inverse (or transpose) of the rotation matrix, ensuring that transformation is invertible.
- This technique is used in various fields such as physics, computer graphics, and engineering to simplify problems and align systems along more convenient axes.
