Question

A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) clockwise rotation by an angle of \(90^{\circ}\) (b) horizontal reflection across the \(y\) axis (c) vertical shear by a factor of 1

Short answer

The transformation matrix is \( A = \begin{bmatrix} 0 & -1 \\ -1 & -1 \end{bmatrix} \).

Step-by-step solution

Identify Rotation Matrix

A rotation of 90 degrees clockwise on the Cartesian plane can be described by the matrix transformation: \[ R = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \] This matrix swaps the coordinates and changes the sign of the x-coordinate, effectively rotating the point 90 degrees clockwise.

Identify Horizontal Reflection Matrix

A reflection across the y-axis is described by the matrix transformation: \[ H = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \] This matrix changes the sign of the x-coordinate, reflecting the point across the y-axis.

Identify Vertical Shear Matrix

A vertical shear by a factor of 1 is described by the matrix transformation: \[ S = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \] This matrix adds the x-coordinate to the y-coordinate, shearing the point vertically by a factor of 1.

Combine Transformations

The combined transformation matrix that applies these transformations in the given order (a), (b), and (c) is obtained by multiplying these matrices: \[ A = S \times H \times R \] First, compute the product of H and R: \[ HR = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \] Then, multiply S and HR: \[ SH = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \ -1 & -1 \end{bmatrix} \] Thus, the matrix \( A \) that represents the sequence of transformations is: \[ A = \begin{bmatrix} 0 & -1 \ -1 & -1 \end{bmatrix} \]

Key concepts

Rotation Matrix

The rotation matrix is a crucial tool for rotating points around the origin on a Cartesian plane. In our specific example, we are dealing with a 90-degree clockwise rotation. To perform this, we use the following rotation matrix: \[ R = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \] This matrix effectively swaps the x and y coordinates. Additionally, it changes the sign of what was previously the x-coordinate, transforming a point by 90 degrees clockwise.

• Elements of the matrix express how each coordinate contributes to the new coordinates.
• The first column acts on the x-coordinate, while the second column affects the y-coordinate.

This change takes any point on the plane and rotates it around the origin by 90 degrees, creating new coordinates shifted from the original position.

Reflection Matrix

Reflection matrices are used to 'mirror' points over a specific line or axis. In this case, a horizontal reflection across the y-axis is performed using the matrix: \[ H = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \] This specific matrix alters the sign of the x-coordinate, resulting in a reflection over the y-axis. This transformation keeps the y-coordinate the same, while the x-coordinate is reflected.

• The negative factor in the matrix reflects the x values.
• This transformation negates x, effectively flipping the point's position horizontally.

Reflection matrices are versatile and can be used for different kinds of reflections such as over both axes or other lines, but they always primarily involve sign changes.

Shear Matrix

Shear matrices change the shape of an object along one axis. In the exercise, a vertical shear by a factor of 1 is performed using the matrix: \[ S = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \] This matrix adds the x-coordinate to the y-coordinate. Thus, the y-value of any point is increased by the amount of the x-value, resulting in a 'shearing' effect vertically.

• In a shear matrix, one row remains all ones to keep the axis scaling the same.
• Shear matrices can change either the x or y axis, based on their structure.

In our example, the vertical shear means movement or distortion is parallel to the y-axis, creating a slant of the geometric figure.