A vector carries both a direction and a size, written as an x part and a y part.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Adding
Add the x parts and the y parts separately. Nothing mixes across.
Why that works
Drawn head to tail, the two vectors and their sum close a triangle.
Magnitude
The length is the hypotenuse of the components, by Pythagoras.
Direction and distance in one object
A vector has magnitude and direction but no fixed position. The vector (3, 4) means "3 right and 4 up" from wherever you start. That positionlessness is what distinguishes a vector from a point.
Adding is following one then the other
Vectors add component by component, which geometrically means placing them nose to tail. The sum is the single displacement equivalent to making both moves in sequence.
Magnitude uses Pythagoras
The length of (3, 4) is √(9 + 16) = 5. Magnitude is distance from the origin to the point with those coordinates, so the distance formula computes it directly.
Why vectors are worth introducing
Forces, velocities, and displacements all combine as vectors. Physics is unusable without them, and computer graphics represents every position and motion this way. The plane is where the idea is easiest to see.
Step 2: Try It Yourself
Move both vectors and watch the sum close the triangle.
- Vector a(3, 4) · length 5
- Vector b(2, -1) · length 2.24
- a + b(5, 3) · length 5.83
The dashed arrow is b again, moved to the tip of a. The sum closes the triangle, and its components are just the x parts added and the y parts added.
Step 3: Watch an Example
One step at a time.
Watch Kofi Add and Measure
Kofi adds (3, 4) and (2, −1), then measures the first.
- Step 1
The x parts add: 3 + 2 = 5.
Step 4: Your Turn
Practice makes it stick.
The Sum
Problem 1 of 2
(4, 1) + (2, 6). What is the y component?
The Length
Problem 2 of 2
Magnitude of (9, 12)?
Components and Length
1 of 8
(1, 3) + (5, 2). x component?
2 of 8
Same sum. y component?
3 of 8
Magnitude of (6, 8)?
4 of 8
Magnitude of (5, 12)?
5 of 8
(4, 5) + (−4, −5). x component?
6 of 8
3 times (2, 4). y component?
7 of 8
Magnitude of (0, 6)?
8 of 8
Do the x and y parts mix when adding vectors? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Magnitude of (8, 15)?
What You Learned
- A vector is an x part and a y part together.
- Vectors add component by component, and the picture closes a triangle.
- The magnitude is the hypotenuse of the components.