Understanding a 3 by 3 projection matrix

🎥 Understanding a 3 by 3 Projection Matrix - A Beginner's Guide 🤔📐

In this video, we will explore the concept of a projection matrix and how it is used to project points from three-dimensional space onto a two-dimensional plane. 🌎

We will start by imagining an axis coming off one direction, another axis going in another direction, and another axis going in the third direction. Then, we will discuss how to extract the x sub 1 and x sub 2 coordinates while leaving out the x sub 3 coordinate, which is the third three-dimensional coordinate.

To accomplish this mathematically, we will use a 3 by 3 matrix where the first column will contain 1, 0, and 0. The second column will contain 0, 1, and 0. The third column will contain 0, 0, and 0.

We will then multiply this matrix by the vector representation of the point, which is x sub 1, followed by x sub 2, followed by x sub 3. By carrying out this multiplication, we will obtain a 3 by 1 column vector, which we will add to get the final projection of the point onto the two-dimensional plane.

Through the use of detailed examples and mathematical notation, we will show you how this process works step-by-step. You will learn how to distribute x sub 1 and x sub 2, multiply them by the unit vectors along the corresponding axes, and then add the resulting scaled vectors to get the projected point in two-dimensional space.

If you're interested in learning about projection matrices and how they work, this video is perfect for you! Don't forget to like and subscribe for more informative content.

🤔📐 Common Questions Answered in this Video 🎥

1️⃣ What is a projection matrix?
A projection matrix is a matrix used to project points from three-dimensional space onto a two-dimensional plane.

2️⃣ How do you extract the x sub 1 and x sub 2 coordinates from a point in three-dimensional space?
To extract the x sub 1 and x sub 2 coordinates from a point in three-dimensional space, you can use a projection matrix.

3️⃣ What is the size of the projection matrix used in this video?
The projection matrix used in this video is a 3 by 3 matrix.

4️⃣ What is the purpose of a projection matrix?
The purpose of a projection matrix is to project points from three-dimensional space onto a two-dimensional plane.

5️⃣ How do you multiply a projection matrix and a point in three-dimensional space?
To multiply a projection matrix and a point in three-dimensional space, you can use matrix multiplication.

6️⃣ How do you project a point from three-dimensional space onto a two-dimensional plane?
To project a point from three-dimensional space onto a two-dimensional plane, you can use a projection matrix.

7️⃣ What are the unit vectors used in the projection matrix?
The unit vectors used in the projection matrix are (1, 0, 0) for the x-axis, (0, 1, 0) for the y-axis, and (0, 0, 0) for the z-axis.

8️⃣ What is the resulting vector after multiplying the projection matrix and the point in three-dimensional space?
The resulting vector after multiplying the projection matrix and the point in three-dimensional space is a 3 by 1 column vector.

9️⃣ How do you distribute x sub 1 and x sub 2 when multiplying the projection matrix and the point in three-dimensional space?
To distribute x sub 1 and