What is a Vector in Math?

A vector is just an ordered list of numbers that helps describe something in space. These numbers (also called components) tell us information like position, direction, or size (magnitude).

Think of a vector as an arrow pointing somewhere. The numbers in the vector tell us where the arrow is pointing and how far it goes.

Imagine you are playing a game on a chessboard. Each square on the board has a position.

If you start at square (2,3) and move 3 steps right and 2 steps up, your new position is:

(2+3, 3+2) = (5,5)

The movement you made can be written as a vector:

(3,2)

This means:

• 3 steps in the x-direction (right)
• 2 steps in the y-direction (up)

So, the vector (3,2) describes how you moved!

Now, think about an airplane flying in the sky. The plane moves in three directions:

1. Forward/backward (x-axis)
2. Left/right (y-axis)
3. Up/down (z-axis)

If the airplane moves:

• 10 km forward
• 5 km to the right
• 2 km up

We write this as a 3D vector:

(10, 5, 2)

This means:

• Move 10 in the x-direction (forward)
• Move 5 in the y-direction (right)
• Move 2 in the z-direction (up)

This vector tells us exactly how the airplane moved in space!

Sometimes, things have more than three important numbers.

For example, if you are describing a person's profile in a system, a vector might look like this:

(25, 180, 75, 4)

Where:

• 25 = Age
• 180 = Height in cm
• 75 = Weight in kg
• 4 = Number of hours they exercise per week

Even though this is not about position or movement, it is still a vector because it is an ordered list of numbers that represents something.

Vectors help us:

• ✔ Describe things using numbers
• ✔ Compare things easily (e.g., finding similar profiles)
• ✔ Measure distance between objects (e.g., in a vector database, we can find the closest match)

A vector is just a list of numbers.

It can represent movement, position, or even other types of information.

• 2D vector → (x, y) for flat surfaces
• 3D vector → (x, y, z) for real-world space
• Higher-dimensional vectors → Used in AI, databases, and more