What Are Vectors?

Vectors are mathematical objects that have both direction and magnitude. You can imagine a vector as an arrow pointing from one place to another in space. In math, we often write vectors as ordered lists of numbers, like \( \mathbf{v} = (2, 3) \) in two dimensions.

Exploring Vector Spaces

A vector space is a collection of vectors that you can add together and multiply by numbers (called scalars) to get new vectors. For a set to be a vector space, it must follow certain rules, such as:

• Adding any two vectors in the space gives another vector in the space.
• Multiplying any vector by a scalar (like 2 or -1) keeps it in the space.

Why Do We Care About Vectors?

Vectors are everywhere! They describe movement, forces, and even data in computer science.

Real-World Connections

• Physics uses vectors to describe velocity and force.
• Computer graphics use vectors to move and rotate images.

Operations on Vectors

• Addition: \( (1,2) + (3,4) = (4,6) \)
• Scalar Multiplication: \( 2 \times (1,2) = (2,4) \)

Working with vectors is like working with arrows—combine them, stretch them, or shrink them!