Vectors are the building blocks of linear algebra. But what exactly is a vector?
Different people see vectors differently. Let's explore each perspective.
Three Ways to See Vectors
Physics Perspective
A vector is an arrow pointing in space.
It has two properties:
- Length - How long it is
- Direction - Where it points
You can move the arrow anywhere in space. As long as the length and direction stay the same, it's the same vector.
Computer Science Perspective
A vector is an ordered list of numbers.
Think of modeling house prices:
- First number: square footage
- Second number: price
The order matters. [1500, 300000] is different from [300000, 1500].
The length of the list determines the dimension. Two numbers = 2D vector. Three numbers = 3D vector.
Mathematician's Perspective
A vector is anything you can add together and multiply by a number.
This sounds abstract. But it highlights the two most important operations in linear algebra:
- Vector addition
- Scalar multiplication
Thinking About Vectors
For learning linear algebra, think of a vector as an arrow rooted at the origin.
Picture the XY plane. The origin (0, 0) is where all vectors start. The arrow points to somewhere in space.
Coordinates
Two Dimensions
The XY plane has:
- X-axis - Horizontal line
- Y-axis - Vertical line
- Origin - Where they meet (0, 0)
A vector's coordinates tell you how to reach its tip from the origin.
Example: Vector [3, 2]
- Move 3 units right (x direction)
- Move 2 units up (y direction)
The first number = horizontal movement
The second number = vertical movement
Positive goes right/up. Negative goes left/down.
Vectors are written vertically in brackets:
[3]
[2]Three Dimensions
Add a third axis:
- Z-axis - Perpendicular to both X and Y
Now vectors have three numbers: [x, y, z]
Each number tells you how far to move along that axis.
Vector Operations
Two fundamental operations make linear algebra work.
1. Vector Addition
Geometrically
Place the tail of the second vector at the tip of the first. Draw a new arrow from the start to the end. That's your sum.
Think of it as combining movements. Walk east, then north. Your total displacement is the sum of both movements.
Numerically
Add the matching components.
[x1] [x2] [x1 + x2]
[y1] + [y2] = [y1 + y2]Example:
[3] [1] [4]
[2] + [5] = [7]2. Scalar Multiplication
A scalar is just a fancy word for a regular number.
Geometrically
Multiplying by a scalar stretches or squishes the vector.
- Multiply by 2 → Vector becomes twice as long
- Multiply by 0.5 → Vector becomes half as long
- Multiply by -1 → Vector flips direction
This is why we call it "scaling."
Numerically
Multiply each component by the scalar.
[x] [c·x]
c · [y] = [c·y]Example:
[3] [6]
2 · [2] = [4]Why This Matters
The power of linear algebra comes from switching between two views:
- Geometric - Arrows in space
- Numeric - Lists of numbers
Data analysts use the geometric view to visualize patterns.
Physicists and game developers use the numeric view to describe and manipulate space with code.
Both views describe the same thing. Learn to switch between them.
Key Takeaways
- Vector = Arrow from origin OR ordered list of numbers
- Coordinates = How to reach the tip from origin
- Addition = Combine movements / add matching components
- Scalar multiplication = Scale length / multiply each component