Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What transformation does a rotation matrix utilize?

1. Addition of angles

2. Multiplication by cosθ and sinθ

3. Subtraction of coordinates

4. Division by scalar values

The correct answer is: Multiplication by cosθ and sinθ

A rotation matrix fundamentally employs the principles of trigonometry to achieve rotation transformations in a coordinate system, specifically utilizing the cosine and sine functions. When rotating a point in a two-dimensional plane around the origin, the new coordinates of the point are derived by multiplying the original coordinates by a matrix that includes cosθ and sinθ. The rotation matrix for an angle θ is structured as follows: \[ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \] This matrix effectively combines the original coordinates (x, y) with the cosine and sine of the angle θ, allowing it to transform the coordinates into their new positions after the specified rotation. Thus, the correct operation that a rotation matrix utilizes is indeed the multiplication by cosθ and sinθ, leading to the new position of the point based on the angle of rotation. In contrast, other options do not accurately describe the transformation performed by a rotation matrix. Adding angles or subtracting coordinates does not adhere to the mathematical principles involved in rotation, and division by scalar values is unrelated to the rotational transformation process.