Find the matrix that produces the given rotation. 45° about the z-axis

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 39E
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Find the matrix that produces the given rotation.

**Topic: Rotation Matrix in 3D Space**

**Objective:** Find the matrix that produces the given rotation: 45° about the z-axis.

**Explanation:**

This exercise involves creating a rotation matrix for a specific rotation in three-dimensional space. The task is to find the matrix capable of rotating a vector 45 degrees around the z-axis. 

**Rotation Matrix Formula:**

A rotation about the z-axis by an angle θ is represented by the matrix:

\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

For a 45° (π/4 radians) rotation, the cosine and sine values are:

- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)

Therefore, the rotation matrix becomes:

\[
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

**Conclusion:** 

Using these values, fill in the matrix to complete the rotation transformation for a 45° rotation about the z-axis.
Transcribed Image Text:**Topic: Rotation Matrix in 3D Space** **Objective:** Find the matrix that produces the given rotation: 45° about the z-axis. **Explanation:** This exercise involves creating a rotation matrix for a specific rotation in three-dimensional space. The task is to find the matrix capable of rotating a vector 45 degrees around the z-axis. **Rotation Matrix Formula:** A rotation about the z-axis by an angle θ is represented by the matrix: \[ \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} \] For a 45° (π/4 radians) rotation, the cosine and sine values are: - \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\) - \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\) Therefore, the rotation matrix becomes: \[ \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix} \] **Conclusion:** Using these values, fill in the matrix to complete the rotation transformation for a 45° rotation about the z-axis.
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