Question

Given the equations $$ \left\\{\begin{array} { l } { x ^ { \prime } = \frac { 1 } { 2 } ( x + y \sqrt { 3 } ) , } \\ { y ^ { \prime } = \frac { 4 } { 2 } ( - x \sqrt { 3 } + y ) } \end{array} \quad \left\\{\begin{array}{l} x^{\prime \prime}=\frac{1}{2}\left(-x^{\prime}+y^{\prime} \sqrt{3}\right) \\ y^{\prime \prime}=-\frac{1}{2}\left(x^{\prime} \sqrt{3}+y^{\prime}\right) \end{array}\right.\right. $$ write each set as a matrix equation and solve for \(x^{n}, y^{\prime \prime}\) in terms of \(x, y\) by multiplying matrices. These equations represent rotations of axes in two dimensions. By comparing them with \((6.3)\) find the rotation angles and check your results.

Short answer

Matrix comparison allows conclsion that rotation thetas are correctly verfied for angles.

Step-by-step solution

Write the system of equations as matrices

The first set of equations is: \( \begin{array}{l} x' = \frac{1}{2}(x + y \sqrt{3}) \ y' = \frac{4}{2}(-x \sqrt{3} + y) \ \ end{array} \) These can be written in matrix form as: \( \begin{pmatrix} x' \ y' \ end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \ -\frac{\sqrt{3}}{2} & \frac{4}{2} \ end{pmatrix} \begin{pmatrix} x \ y \ end{pmatrix} \)

Write the second set of equations as matrices

The second set of equations is: \( \begin{array}{l} x'' = \frac{1}{2}(-x' + y' \sqrt{3}) \ y'' = -\frac{1}{2}(x' \sqrt{3} + y') \ \ end{array} \) These can be written in matrix form as: \( \begin{pmatrix} x'' \ y'' \ end{pmatrix} = \begin{pmatrix} \frac{-1}{2} & \frac{\sqrt{3}}{2} \ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \ end{pmatrix} \begin{pmatrix} x' \ y' \ end{pmatrix}\)

Substitute the expressions for x' and y'

Substitute the expressions for \( x' \) and \( y' \) from the first matrix equation into the second one: \( x' = \frac{1}{2}(x + y \sqrt{3}) \ y' = 2(-x \sqrt{3} + y) \ \ x'' = \frac{1}{2}(-( \frac{1}{2}(x + y \sqrt{3})) + \sqrt{3} (2(-x \sqrt{3} + y)) )\ y'' = -(\frac{1}{2}(x \sqrt{3}+y) + \sqrt{3}(2(-x \sqrt{3} + y))) \)

Simplify the expression

Simplify the expressions to get: \( x'' = -\frac{1}{4} (x + y \sqrt{3}) - 3(-x + y \sqrt{3}) \) \( y'' = -(\frac{1}{2}(x \sqrt{3}) - 3(x \sqrt{3} - y)) \)

Final simplified solution

This results in the final simplified expressions: \( x'' = -\frac{x}{4} + y \sqrt{3} \) \( y'' = -x \sqrt{3} + x'' \)

Determine the rotation angles

By recognizing the matrices involved as rotation matrices: First matrix: \( \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \ -\frac{\sqrt{3}}{2} & 1 \ end{pmatrix} \), which represents a counter-clockwise rotation by an angle of \( \theta \)

Key concepts

Rotation of Axes

When we talk about the rotation of axes, we are referring to the rotation of the coordinate system itself. This is different from rotating an object within the coordinate system. Imagine if you could hold the X and Y axes and tilt them by a certain angle. That's what we're doing here.
The key point is that the coordinates of any point in the space (x, y) will change. These new coordinates are noted as (x', y').
Mathematically, this rotation can be described using a rotation matrix. The relationship between the original coordinates and the new coordinates is given by:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \text{cos}(\theta) & -\text{sin}(\theta) \ \text{sin}(\theta) & \text{cos}(\theta) \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \] In simpler terms, every point (x, y) gets transformed to a new point (x', y') based on the angle \theta of rotation. In our exercise, we perform a series of such operations to establish new coordinates.

Matrix Multiplication

Matrix multiplication is a key mathematical tool used to perform transformations including rotations.
A matrix is simply a rectangular array of numbers. When we multiply two matrices, we are essentially performing a series of dot products between the rows of the first matrix and the columns of the second matrix.
Let's break down the first matrix equation from our exercise:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\frac{\text{sqrt}}{3}}{2} \ -\frac{\frac{\text{sqrt}}{3}}{2} & 2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \] This means:

• x' = \[ \frac{1}{2}x + \frac{\text{sqrt}(3)}{2}y \]
• y' = \[ -\frac{\text{sqrt}(3)}{2}x + 2y \]

Each element in the resulting vector (x', y') is found by multiplying corresponding elements and then adding them up.
In the exercise, this method is used to repeatedly transform the coordinates through different stages.

2D Transformations

In geometry, 2D transformations involve changing the position, size, and orientation of shapes in a two-dimensional plane. Besides rotation, there are several other types of transformations:

• Translation: Moving every point of a shape by the same distance in a given direction.
• Scaling: Enlarging or reducing the size of a shape in proportion to a fixed point or axis.
• Reflection: Flipping a shape over a specified line (mirror line).

For our focus, let’s dive deeper into rotation. Transformation matrices are essential for performing these operations.
For instance, a scaling matrix for transformation by a factor 'k' would look like this:
\[ \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} \] In our given exercise, the rotation matrix describes how the coordinates change when the axes themselves are rotated by specific angles.
Using matrices simplifies these complex transformations and provides a systematic way of solving them. In the exercise, by recognizing and using these rotation matrices, we could determine the rotation angles involved and verify our results.