Problem 36 - Algebra 2

Question

GEOMETRY The matrix used to rotate a figure \(270^{\circ}\) counterclockwise about the origin is \(\left[\begin{array}{cc}{0} & {1} \\ {-1} & {0}\end{array}\right]\) Compare this matrix with the matrix used to rotate figure \(90^{\circ}\) counterclockwise about the origin. a. Are they inverses? b. Does your answer make sense? Use a drawing to support your answer.

Step-by-Step Solution

Answer

The matrices are inverses; applying both rotations results in the original position.

Step 1: Write the Rotation Matrix for 90° Counterclockwise

The standard matrix to rotate a point or figure 90° counterclockwise about the origin is \( \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \).

Step 2: Calculate the Product of the Two Matrices

To determine if the matrices are inverses, multiply the 270° rotation matrix \( \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) by the 90° rotation matrix \( \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \). The resulting product is computed as follows:\[\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} (0)(0) + (1)(1) & (0)(-1) + (1)(0) \ (-1)(0) + (0)(1) & (-1)(-1) + (0)(0) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]A product of \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \) indicates the identity matrix, suggesting they are inverses.

Step 3: Interpret the Result

Since the product of the 270° and 90° rotation matrices is the identity matrix, these matrices are inverses of each other. This means applying one rotation followed by the other brings any point back to its original position.

Step 4: Support with a Drawing

Visually demonstrate the rotations. Draw a coordinate system and place a clear initial point. Show the point's location after a 270° counterclockwise rotation and then a 90° counterclockwise rotation. The point should return to its initial location, confirming that the two matrices are indeed inverses geometrically.

Key Concepts

Inverse Matrices

In mathematics, inverse matrices are a pair of matrices that, when multiplied together, yield the identity matrix. The identity matrix is like a "do-nothing" transformation in matrix operations. It is very similar to multiplying a number by one, leaving the number unchanged. When two matrices are inverses, multiplying one by the other effectively "undoes" the transformations applied by each matrix.

In the context of the given problem, the rotation matrices for 90° and 270° are inverses, since their product is the identity matrix:

• The 90° rotation matrix: \(\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}\)
• The 270° rotation matrix:\(\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}\)

By calculating their product, you can see the result is indeed the identity matrix, confirming their inverse relationship. Understanding inverse matrices helps in reversing transformations, which is crucial in solving many geometric and algebraic problems.

Rotation Matrices

Rotation matrices are fundamental when it comes to transforming coordinates in a plane, typically rotating points around the origin. A rotation matrix is structured to shift points around the origin by a specific angle without altering their distance from the origin.

For example, a counterclockwise rotation through 90°: \[\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}\]and 270°:\[\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}\]Reflect how these matrices organize their numbers to achieve their specific transformations. Rotation matrices are powerful tools used not only in pure mathematics but also in computer graphics, robotics, and engineering. Each matrix preserves the length of the vector being rotated, maintaining the integrity of shapes and their dimensions.

Identity Matrix

The identity matrix is a special type of matrix that acts as the multiplicative "neutral" element in matrix operations. Just like multiplying a number by one doesn’t change the number, multiplying a matrix by the identity matrix doesn’t alter the original matrix. The identity matrix for a 2x2 system looks like this:\[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

It works as if saying, "I've done nothing to this matrix." When you multiply a transformation matrix by its inverse, the outcome is the identity matrix, confirming their inverse nature. This principle proves invaluable in geometry and algebra because it allows transformations to be undone, ensuring that the original data or shape can be recovered after transformations.

Counterclockwise Rotation

A counterclockwise rotation involves turning a vector or shape about the origin, in a direction opposite to the movement of the hands of a clock. This type of rotation is common in standard mathematical and geometric transformations. For instance, a 90° counterclockwise rotation twists the vectors by a quarter turn to the left from their starting positions.

In the given problem:

• The 90° rotation matrix transforms a vector point by 90° counterclockwise.
• The 270° rotation matrix achieves what a 90° clockwise rotation would, due to the cumulative effect.

Visualizing these rotations on a coordinate grid provides insight into operation of matrices. Rotations, especially counterclockwise, are vital in graphics, animations, and design modeling, aiding in how objects and shapes are manipulated within a specified space. Understanding these can enhance spatial reasoning and provide clarity on how geometric transformations apply practically.