Problem 30 - Algebra 2

Question

For Exercises \(28-31,\) use rectangle \(A B C D\) with vertices \(A(-4,4), B(4,4),\) \(C(4,-4),\) and \(D(-4,-4) .\) Find the coordinates of the image in matrix form after a reflection over the line \(y=x .\)

Step-by-Step Solution

Answer

The image coordinates after reflection are: A'(4, -4), B'(4, 4), C'(-4, 4), D'(-4, -4).

Step 1: Understand Reflection Matrix

To reflect over the line \(y = x\), the reflection matrix is \(\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\). This matrix will swap the x and y coordinates of any point.

Step 2: Represent Vertices as Matrices

Express the points \(A(-4,4), B(4,4), C(4,-4), D(-4,-4)\) as column matrices: \(\begin{bmatrix} -4 \ 4 \end{bmatrix}, \begin{bmatrix} 4 \ 4 \end{bmatrix}, \begin{bmatrix} 4 \ -4 \end{bmatrix}, \begin{bmatrix} -4 \ -4 \end{bmatrix}\).

Step 3: Apply the Reflection Matrix

Multiply each vertex matrix by the reflection matrix \(\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\). For example, for point \(A\), compute: \[ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} -4 \ 4 \end{bmatrix} = \begin{bmatrix} 4 \ -4 \end{bmatrix} \]Do this for all vertices.

Step 4: Calculate Image for Each Vertex

For point \(A(-4, 4)\), the image is \((4, -4)\).For point \(B(4, 4)\), the image is \((4, 4)\).For point \(C(4, -4)\), the image is \((-4, 4)\).For point \(D(-4, -4)\), the image is \((-4, -4)\).

Step 5: Write the Final Coordinates

After performing the matrix operations, the coordinates of the reflected image are: - Image of \(A\): (4, -4)- Image of \(B\): (4, 4)- Image of \(C\): (-4, 4)- Image of \(D\): (-4, -4).

Key Concepts

Reflection Matrix

A reflection matrix is a special tool in mathematics used to reflect points around a line in the coordinate plane. For reflections over the line \(y = x\), we use the matrix \(\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\). This matrix essentially swaps the x-coordinate with the y-coordinate of a given point. When you multiply a point's coordinates by this reflection matrix, you effectively mirror that point over the line \(y = x\). This is a neat way of transforming geometric figures and is often used in various fields, including computer graphics and robotics.

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra, involving the multiplication of two matrices to produce another matrix. When we talk about multiplying a point by a reflection matrix, we treat the point as a column matrix. For instance, a point \((x, y)\) is represented as \(\begin{bmatrix} x \ y \end{bmatrix}\). During multiplication, the reflection matrix \(\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\) operates on this column matrix.The process involves the first row of the reflection matrix being multiplied with the point's column to form the first entry for the new matrix. Similarly, the second row of the reflection matrix is multiplied with the point's column to form the second entry. The result is a new matrix that represents new coordinates. This operation swaps the original \(x\) and \(y\) values.

Coordinate Geometry

Coordinate geometry, or analytic geometry, deals with points on the coordinate plane and uses algebra to define geometrical concepts. In this context, we have a rectangle defined by its vertices: \(A(-4,4), B(4,4), C(4,-4),\) and \(D(-4,-4)\). Each vertex is a point placed on the Cartesian plane, where the x-axis and y-axis intersect.The beauty of coordinate geometry is that it allows us to see the effects of transformations, like reflections, in a clear manner. By applying reflection matrices to these points, we can find their new positions after reflection. This technique helps in visualizing and evaluating geometric transformations in a structured way, making complex problems easier to solve and understand.

Transformation

Transformations in geometry involve changing the position or orientation of a shape. These can include translations, rotations, dilations, and reflections. Specifically, reflection is a flip over a certain line, creating a mirror image of a shape. In the given problem, reflecting a rectangle over the line \(y = x\) gives us a new shape with its points positioned differently.By using mathematical tools like matrices, transformations become more manageable. Applying the reflection matrix to each vertex of the rectangle, the points are transformed or 'flipped.' Consequently, understanding these transformations is crucial for solving problems involving physical movements in space and is widely used in technology sectors such as animation and engineering.