Plotting 3x3 Rotational Matrices And Improper Rotations

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Introduction

Rotational matrices are a fundamental concept in linear algebra and geometry, used to describe the rotation of objects in 3D space. However, visualizing these matrices can be a challenging task, especially for students who are new to the subject. In this article, we will explore how to plot 3x3 rotational matrices and improper rotations using Desmos, a popular online graphing calculator.

What are Rotational Matrices?

A rotational matrix is a 3x3 matrix that represents a rotation in 3D space. It is used to describe the transformation of a point or a vector from one position to another. The matrix has the following form:

| a  b  c |
| d  e  f |
| g  h  i |

where a, b, c, d, e, f, g, h, and i are the elements of the matrix.

Types of Rotational Matrices

There are two types of rotational matrices: proper and improper. A proper rotational matrix represents a rotation of a point or a vector around an axis, while an improper rotational matrix represents a rotation followed by a reflection.

Proper Rotational Matrices

A proper rotational matrix has the following form:

| cos(θ)  -sin(θ)  0 |
| sin(θ)   cos(θ)  0 |
| 0        0       1 |

where θ is the angle of rotation.

Improper Rotational Matrices

An improper rotational matrix has the following form:

| cos(θ)  -sin(θ)  0 |
| sin(θ)   cos(θ)  0 |
| 0        0       -1 |

Plotting Rotational Matrices in Desmos

To plot a rotational matrix in Desmos, we need to create a 3D graph and use the matrix to transform a point or a vector. Here's an example of how to plot a proper rotational matrix:

Step 1: Create a 3D Graph

To create a 3D graph in Desmos, we need to select the "3D" option from the graph menu.

Step 2: Define the Matrix

We need to define the rotational matrix as a 3x3 matrix. We can do this by using the following code:

A = matrix([[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, 1]]);

Step 3: Define the Point or Vector

We need to define a point or a vector to transform. We can do this by using the following code:

P = vector([1, 0, 0]);

Step 4: Transform the Point or Vector

We need to transform the point or vector using the rotational matrix. We can do this by using the following code:

Q = A * P;

Step 5: Plot the Graph

We need to plot the graph of the transformed point or vector. We can do this by using the following code:

plot3D(Q);

Example: Plotting Proper Rotational Matrix

Here's an example of how to plot a proper rotational matrix in Desmos:

# Proper Rotational Matrix

Step 1: Create a 3D Graph


    

  • Select the "3D" option from the graph menu.
    • 



Step 2: Define the Matrix


A = matrix([[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, 1]]);
</code></pre>
<h2>Step 3: Define the Point or Vector</h2>
<pre><code class="hljs">P = vector([1, 0, 0]);
</code></pre>
<h2>Step 4: Transform the Point or Vector</h2>
<pre><code class="hljs">Q = A * P;
</code></pre>
<h2>Step 5: Plot the Graph</h2>
<pre><code class="hljs">plot3D(Q);
</code></pre>
<h2><strong>Example: Plotting an Improper Rotational Matrix</strong></h2>
<p>Here's an example of how to plot an improper rotational matrix in Desmos:</p>
<pre><code class="hljs"># Improper Rotational Matrix

## Step 1: Create a 3D Graph

* Select the &quot;3D&quot; option from the graph menu.

## Step 2: Define the Matrix

```math
A = matrix([[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, -1]]);
</code></pre>
<h2>Step 3: Define the Point or Vector</h2>
<pre><code class="hljs">P = vector([1, 0, 0]);
</code></pre>
<h2>Step 4: Transform the Point or Vector</h2>
<pre><code class="hljs">Q = A * P;
</code></pre>
<h2>Step 5: Plot the Graph</h2>
<pre><code class="hljs">plot3D(Q);
</code></pre>
<h2><strong>Conclusion</strong></h2>


<h2><strong>Frequently Asked Questions</strong></h2>
<h2><strong>Q: What is a rotational matrix?</strong></h2>
<p>A: A rotational matrix is a 3x3 matrix that represents a rotation in 3D space. It is used to describe the transformation of a point or a vector from one position to another.</p>
<h2><strong>Q: What are the types of rotational matrices?</strong></h2>
<p>A: There are two types of rotational matrices: proper and improper. A proper rotational matrix represents a rotation of a point or a vector around an axis, while an improper rotational matrix represents a rotation followed by a reflection.</p>
<h2><strong>Q: How do I plot a rotational matrix in Desmos?</strong></h2>
<p>A: To plot a rotational matrix in Desmos, you need to create a 3D graph and use the matrix to transform a point or a vector. Here's an example of how to plot a proper rotational matrix:</p>
<h3>Step 1: Create a 3D Graph</h3>
<ul>
<li>Select the &quot;3D&quot; option from the graph menu.</li>
</ul>
<h3>Step 2: Define the Matrix</h3>
<pre><code class="hljs">A = matrix([[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, 1]]);
</code></pre>
<h3>Step 3: Define the Point or Vector</h3>
<pre><code class="hljs">P = vector([1, 0, 0]);
</code></pre>
<h3>Step 4: Transform the Point or Vector</h3>
<pre><code class="hljs">Q = A * P;
</code></pre>
<h3>Step 5: Plot the Graph</h3>
<pre><code class="hljs">plot3D(Q);
</code></pre>
<h2><strong>Q: How do I plot an improper rotational matrix in Desmos?</strong></h2>
<p>A: To plot an improper rotational matrix in Desmos, you need to create a 3D graph and use the matrix to transform a point or a vector. Here's an example of how to plot an improper rotational matrix:</p>
<h3>Step 1: Create a 3D Graph</h3>
<ul>
<li>Select the &quot;3D&quot; option from the graph menu.</li>
</ul>
<h3>Step 2: Define the Matrix</h3>
<pre><code class="hljs">A = matrix([[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, -1]]);
</code></pre>
<h3>Step 3: Define the Point or Vector</h3>
<pre><code class="hljs">P = vector([1, 0, 0]);
</code></pre>
<h3>Step 4: Transform the Point or Vector</h3>
<pre><code class="hljs">Q = A * P;
</code></pre>
<h3>Step 5: Plot the Graph</h3>
<pre><code class="hljs">plot3D(Q);
</code></pre>
<h2><strong>Q: What is the difference between a proper and an improper rotational matrix?</strong></h2>
<p>A: A proper rotational matrix represents a rotation of a point or a vector around an axis, while an improper rotational matrix represents a rotation followed by a reflection.</p>
<h2><strong>Q: How do I determine the angle of rotation?</strong></h2>
<p>A: The angle of rotation is determined by the value of θ in the rotational matrix. You can adjust the value of θ to change the angle of rotation.</p>
<h2><strong>Q: Can I plot multiple rotational matrices in Desmos?</strong></h2>
<p>A: Yes, you can plot multiple rotational matrices in Desmos by creating multiple 3D graphs and using the matrices to transform points or vectors.</p>
<h2><strong>Q: How do I save my plot in Desmos?</strong></h2>
<p>A: To save your plot in Desmos, you can click on the &quot;Save&quot; button in the top right corner of the graph. This will save the plot as a PNG file.</p>
<h2><strong>Conclusion</strong></h2>
<p>In this article, we have answered some of the most frequently asked questions about plotting 3x3 rotational matrices and improper rotations in Desmos. We have seen how to create a 3D graph, define the matrix, define the point or vector, transform the point or vector, and plot the graph. We have also seen examples of how to plot a proper rotational matrix and an improper rotational matrix. By following these steps, you can gain a better understanding of rotational matrices and improper rotations.</p>