Math 255- Math Fundamentals for Game

What is Math Fundamentals for Game?

Game mathematics turns positions, directions, forces, and shapes into calculations a computer can use. Vectors describe where objects are and where they move; trigonometry connects angles with directions; physics equations update movement over time; and collision tests help determine when objects touch or overlap.

This introduction walks through core ideas step by step, using simple numbers you can later translate into code.

A useful approach: Draw the situation → name the vectors → choose the formula → substitute numbers → check units and sign → test against the game (does it look right?).

1. Vectors

A vector stores a magnitude and a direction as components. In a game it can be a position, a velocity, or a direction.

v = (v_{x}, v_{y}, v_{z})

Core Vector Operations

Operation	Formula	Game Use
Addition	$a + b = (a_{x} + b_{x}, a_{y} + b_{y})$	Combine movement and wind
Scaling	$k v = (k v_{x}, k v_{y})$	Change speed
Magnitude	$\| v \| = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}$	Speed or distance
Normalize	$\hat{v} = \frac{v}{\| v \|}$	Direction only, length 1
Move	$p_{n e w} = p + v Δ t$	Update position each frame

Example: Normalize

v = (3, 4)

Step 1: Find the magnitude.

| v | = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5

Step 2: Divide each component by 5.

\hat{v} = (\frac{3}{5}, \frac{4}{5}) = (0.6, 0.8)

Step 3: Check: $\sqrt{0.36 + 0.64} = 1$ . Conclusion: The unit direction is $(0.6, 0.8)$ .

Example: Normalize

v = (3, 4)

Step 1: Find the magnitude.

| v | = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5

Step 2: Divide each component by 5.

\hat{v} = (\frac{3}{5}, \frac{4}{5}) = (0.6, 0.8)

Step 3: Check: $\sqrt{0.36 + 0.64} = 1$ . Conclusion: The unit direction is $(0.6, 0.8)$ .