Some quantities, like temperature or speed, can be described with a single number. These are called scalars. A vector is different. It's a mathematical object that has both a magnitude (or length) and a direction. Think of it as an arrow pointing in space.

Imagine telling someone how to get from their house to the library. You wouldn't just say "go 2 miles." You'd say "go 2 miles east." That instruction—"2 miles east"—is a vector. It has a magnitude (2 miles) and a direction (east). Velocity is another great example. A car's speed might be 60 miles per hour, but its velocity is 60 miles per hour north.

We often represent vectors using coordinates. In a 2D plane, a vector can be described by two numbers: how far it goes along the x-axis and how far it goes along the y-axis. For example, a vector $\vec{v} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ represents an arrow that moves 3 units to the right and 1 unit up.

We can perform mathematical operations with vectors. The two most basic ones are addition and scalar multiplication.

To add two vectors, we place them tip-to-tail. The new vector, called the resultant, is an arrow drawn from the starting point of the first vector to the ending point of the second. If you walk 3 miles east and then 2 miles north, your final position is the sum of those two vector movements.

Numerically, vector addition is simple. You just add the corresponding components. If $\vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then their sum is:

Scalar multiplication involves changing a vector's length by multiplying it by a scalar (a regular number). Multiplying a vector by 2 doubles its length while keeping its direction. Multiplying by -1 reverses its direction but keeps its length.

To perform scalar multiplication, you multiply each component of the vector by the scalar. If $c$ is a scalar and $\vec{a} = \begin{pmatrix} x \\ y \end{pmatrix}$, then:

When we combine scalar multiplication and vector addition, we create what's called a linear combination. A linear combination of vectors is the sum of those vectors, each scaled by a constant.

For example, given two vectors $\vec{u}$ and $\vec{v}$, a linear combination would look like this:

Here, $c_1$ and $c_2$ are scalars. By choosing different scalars, we can create any other vector in the same plane as $\vec{u}$ and $\vec{v}$, as long as they aren't pointing in the same or opposite directions.

Let's say we have $\vec{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$. We can create a new vector $\vec{w}$ by picking scalars, say $c_1 = 2$ and $c_2 = -3$.

The set of all possible linear combinations of a group of vectors is called their span. If you have two non-parallel vectors in a 2D plane, their span is the entire plane. You can reach any point just by scaling and adding those two vectors.

Understanding how to scale and add vectors is fundamental. These simple operations allow us to navigate and describe complex, multi-dimensional spaces.