Lesson 5 of 22, in world 2: Motion & Math

Vectors

Time: About 20 min
Builds on: Coordinates & Transforms Delta Time & Fixed Timesteps

A vector is just a pair of numbers (x, y). In games, vectors describe positions, directions, velocities, forces and distances — so they’re the single most-used tool in 2D maths. Good news: you only need a handful of operations.

Two ways to read a vector

As a point: “the player is at (300, 200)”.
As an arrow (a direction with a length): “move 3 right and 4 down”, which has length 5.

LiveVector playground

Show

Drag the yellow handles at the tips of A and B. Each grid square is 40 px.

The operations you’ll use every day

Operation	Formula	Game use
Add	`(a.x+b.x, a.y+b.y)`	Apply velocity to position: `pos = pos + vel·dt`
Subtract	`(b.x−a.x, b.y−a.y)`	Direction from enemy to player: `player − enemy`
Scale	`(a.x·s, a.y·s)`	Speed up / slow down a direction
Length	`√(x² + y²)`	Distance between things, current speed
Normalize	`a / length(a)`	Pure direction with length 1
Dot product	`a.x·b.x + a.y·b.y`	”Is the enemy in front of me?” (> 0 means yes)

JavaScript

// Make an enemy chase the player at 120 px/s
const toPlayer = sub(player.pos, enemy.pos)     // arrow from enemy to player
const direction = normalize(toPlayer)           // length 1: just "which way"
enemy.pos = add(enemy.pos, scale(direction, 120 * dt))

// Is the player within 200 px?
if (length(toPlayer) < 200) enemy.alert = true

Normalize your input

Reading arrow keys gives an input vector like (1, 0) for right, or (1, 1) for down-right. But (1, 1) has length √2 ≈ 1.41, so diagonal movement is 41% faster. Normalize the input vector before multiplying by speed:

LiveThe diagonal speed bug

Hold →+↓ together. Both ships move at "200 px/s", but the pink one adds x and y input directly, so diagonally it moves at √2 × 200 ≈ 283 px/s. The cyan one normalizes the input first.

In Godot

Godot has a built-in Vector2 with all of these:

GDScript

var dir = Input.get_vector("ui_left", "ui_right", "ui_up", "ui_down") # already normalized!
velocity = dir * speed
var to_player = player.global_position - global_position
if to_player.length() < 200:
    position += to_player.normalized() * 120 * delta

Your repo’s godot/02-input-map-vectors lesson builds on exactly this.

Key takeaways

A vector is (x, y): a point or an arrow with direction and length.
Subtract to get "from A to B", normalize to get a pure direction, scale to set speed.
Normalize input vectors so diagonal movement isn’t faster.

Check yourself

Q1. Enemy at (100, 100), player at (400, 500). What is the distance between them?

Q2. What does normalize((0, −8)) return?

Q3. To make an enemy FLEE from the player, its direction is…

Exercises

In the playground, make A and B point in opposite directions. What is A + B? When is it (0, 0)?
Compute by hand: dot((1,0), (0,1)), dot((1,0), (1,0)), dot((1,0), (−1,0)). What does the sign tell you?
Pseudo-code: a homing missile that turns toward the player. Which operations do you need?