Wind pushes a sailboat sideways. Gravity pulls a skydiver down. A plane flies east while the jet stream shoves it north. Every one of these is a vector at work — and once you see them, you can't unsee them.
A vector is a quantity that has both a magnitude (how much) and a direction (which way). That's it. But that simple addition — direction — changes everything about how you add, subtract and analyse these quantities compared to ordinary numbers.
This guide walks through what vectors are, how they differ from scalars, how to add and subtract them visually, and how to break them into components. Every concept comes with a diagram you can see and, in two cases, interact with.
Magnitude & Direction
A vector is an arrow: its length is the magnitude, its tip points the direction.
Scalar vs Vector
Scalars have only size. Vectors have size and direction — speed vs velocity, mass vs weight.
Addition & Components
Vectors add tip-to-tail, and any vector can be split into x and y parts.
What Is a Vector, Really?
MAGNITUDE + DIRECTIONThink of a vector as a directed arrow. The arrow's length represents its magnitude — how strong or how large the quantity is. The direction the arrow points represents its direction. Change either one, and you've changed the vector.
In written form, vectors are usually bolded or given an arrow: v or v⃗. Their magnitude is written as |v| or just v, and is always a non-negative number.
For example, a force vector of 10 newtons pointing 30° above the horizontal is a single vector — not two separate things. Its magnitude is 10 N. Its direction is 30° from the x-axis. Both pieces are needed to describe it fully.
Two vectors are equal only if they have the same magnitude and the same direction. A vector pointing north with length 5 is not the same as one pointing east with length 5, even though both have magnitude 5.
Scalar vs Vector: The Critical Difference
ONE NUMBER VS TWO PIECES OF INFORMATIONA scalar is fully described by a single number — its magnitude. Temperature, mass, time, speed, energy and distance are all scalars. You don't need to say "20°C pointing north." Temperature is just 20°C.
A vector needs both magnitude and direction. Velocity, acceleration, force, displacement, momentum and weight are vectors. Saying "50 km/h" is a speed (scalar). Saying "50 km/h due north" is a velocity (vector).
| Scalar | Vector | Why it matters |
|---|---|---|
| Speed | Velocity | Speed tells you how fast; velocity tells you how fast and in which direction. |
| Mass | Weight | Mass is just how much stuff; weight is a force pulling toward Earth's center. |
| Distance | Displacement | Distance is total ground covered; displacement is the straight-line shift from start to end. |
| Energy | Force | Energy is a number; force has a direction and changes motion. |
| Time | Acceleration | Time is a number; acceleration is how velocity changes, including its direction. |
Here's the practical consequence: you can add scalars like ordinary numbers — 5 kg + 3 kg = 8 kg. But you cannot add vectors like that. A 3 N force east plus a 4 N force north is not 7 N. It's 5 N pointing northeast. Direction changes the math.
Vector Addition: Tip-to-Tail
RESULTANT VECTORThe standard way to add two vectors visually is the tip-to-tail method. Draw the first vector. Then place the tail of the second vector at the tip of the first. The resultant — the sum — is the arrow drawn from the tail of the first vector to the tip of the second.
Order doesn't matter. Adding A then B gives the same result as adding B then A. This is called the commutative property, and it works for any number of vectors.
Try it below: drag the sliders to change the x and y components of two vectors, and watch the resultant appear tip-to-tail.
The green arrow is the resultant. Move the sliders and watch how the tip-to-tail construction always produces the sum.
A plane flies east at 200 km/h. Wind blows north at 50 km/h. The plane's actual velocity over the ground is the vector sum of those two — about 206 km/h pointing slightly north of east. This is why flight paths need vector correction.
Vector subtraction
Subtracting a vector is the same as adding its opposite. A − B equals A + (−B), where −B is the same length as B but points exactly the opposite way. So the tip-to-tail method still works — just flip the second vector before you add it.
Components: Breaking Vectors Apart
X AND Y PARTSAny 2D vector can be split into two perpendicular components: a horizontal part (x) and a vertical part (y). This is one of the most useful skills in physics and engineering, because components are just numbers — and numbers are easy to add.
If a vector has magnitude r and points at angle θ from the positive x-axis, its components are:
Going the other way — from components back to magnitude and direction — uses the Pythagorean theorem and inverse tangent:
Components turn vector addition into simple arithmetic. Break each vector into x and y parts, add all the x parts together, add all the y parts together, then rebuild the resultant from those two sums. That's exactly how the interactive diagram above works under the hood.
| Vector | x-component | y-component | Magnitude |
|---|---|---|---|
| A = (3, 4) | 3 | 4 | 5 |
| B = (5, 12) | 5 | 12 | 13 |
| C = (8, 15) | 8 | 15 | 17 |
| D = (1, 1) | 1 | 1 | √2 ≈ 1.414 |
Those first three are Pythagorean triples — right triangles with whole-number sides. They show up constantly in textbook problems because they keep the arithmetic clean.
Where Vectors Show Up in Real Life
Vectors aren't abstract math — they're the language engineers, pilots, game developers and physicists use every day.
- Navigation: GPS and flight planning combine velocity vectors with wind vectors to find the actual path over ground.
- Video games: Every 3D game engine uses vectors to track position, velocity, and the direction a character or camera is facing.
- Structural engineering: Forces on a bridge are vectors — beams must handle both the push (magnitude) and the direction of each load.
- Sports analytics: A soccer ball's trajectory is a vector sum of initial kick velocity and gravity's acceleration.
- Machine learning: Data points are vectors in high-dimensional space; similarity between vectors tells you how alike two items are.
If you walk 3 km east, then 4 km north, you've walked 7 km total — but your displacement from the start is only 5 km, pointing roughly northeast. Distance is scalar addition; displacement is vector addition. They give different answers, and both are correct for different questions.
The Whole Picture in One Diagram
Here's the entire vector toolkit in a single view: a vector, its components, a second vector, and the resultant of adding them tip-to-tail. If you understand this picture, you understand the core of vectors.
Vector v (blue) with its x and y components (dashed). Vector w (amber) added tip-to-tail produces the resultant r (green).