Math

Why I’m Writing This This semester I took a role tutoring MATH14998 (Computer Math) at Sheridan College. I originally felt a little out of water given how long it’s been since last studied math. Throughout the term, I’ve learned a lot about what it takes to teach and some of my short comings and strengths. One of them being, I have a sense of what concepts are particularly difficult grasp and finding ways to explain them so that the students can actually understand the concepts rather than memorize formulas. Now my delivery of the explanations could use some work but I’ve put together some explanations that I’m quite proud of, one of them being combinatorics. And so, I’ve included a compilation of those notes here. ...

Dot Product Find the interior angle between two vectors $$ A\cdot B = |A||B|cos\theta $$ $$ \theta = \arccos(\dfrac{A\cdot B}{|A||B|}) $$ $$ where\ A\cdot B = A_x \times A_y + B_x \times B_y $$ Cross Product Get perpendicular vector from two vectors Find area created by two vectors (for 2D, parallelogram area, for 3D, parallelepiped area) This is the magnitude of perpendicular vector Matrix Multiplication There is a formula for calculating the resultant matrix from a matrix multiplication. ...

What Matters The dot product has a bunch of properties when you use it’s raw scalar output but honestly most of the times you will use the dot product is to find the angle between two vectors by rearranging to equation as follows $$ A\cdot B = |A||B|cos\theta $$ $$ \theta = \arccos(\dfrac{A\cdot B}{|A||B|}) $$ $$ where\ A\cdot B = A_x \times A_y + B_x \times B_y $$ In games, this equation is used exhaustively to calculate light projections and field of view. ...

Two’s Complement Representation Why We Use It In c++, signed integers are represented in two’s complement notation. Before I get to how that notation works, I want to explain why we use it. Comparing both representations, -1 would look like this // Decimal 4294967295 // Binary Signed Int 10000000000000000000000000000001 // Two's Complement Signed Int 11111111111111111111111111111111 The normal signed binary representation is pretty easily understood if you know what a sign bit is. You just have the typical binary for 1 with the most significant bit being the sign bit to indicate that it’s negative. Then why the heck do we use two’s complement instead? Isn’t it just more confusing? ...

I’ve always had trouble understanding probability, especially when entering into the more theoretical aspects of it. Here, I want to cover some of the basic concepts and functions core to probability in an easily digestible format that I can refer to later on when I inevitably forget it all. Random Variable Whenever there’s a question of probability, you tend to have some range of possible outcomes sourced from a specific event. ...

In preparation for a deep learning course I’m taking over the Summer, I’m taking a short intro course on machine learning to help prepare me for some of the fundamental concepts. I’ve been avoiding AI for a while but given its ongoing application in nearly everything now, I figure it’s more than worth getting my feet wet. ML Overview flowchart LR subgraph Shader Lifecycle direction LR d[("Dataset")] --> m(("Model")) --> o("Predicted Values") ---|compare with| a("Actual Values") -->|calculate| l["Loss"] l -->|training data| m end Loss Functions When we compare our predicted results to the actual results, how do we calculate the loss? ...

When describing rotations of an object, the typical way would be through linear algebra and trigonometry. However, similarly to how complex numbers can be used to describe rotations in 2D, quaternions allow efficient and more pragmatic methods describing rotations in 3D. How It’s Done With Trigonometry Depending on the framework or software using, a lot of the work is already done for you with API available to you to perform rotations on shapes. But for a simple point in space rotating in about an origin, what would that look like? ...