Some quantities in the world can be described with a single number. If you say your cat weighs 5 kilograms, that’s the whole story. If the temperature outside is 22° Celsius, we know exactly how warm it is. These are called scalars—quantities that only have magnitude, or size.

But what if you're giving someone directions? Saying "walk 500 meters" isn't enough. You need to specify which way to walk. "Walk 500 meters northeast" is useful information. This combination of a magnitude (500 meters) and a direction (northeast) is a vector.

We represent vectors visually with arrows. The length of the arrow shows the vector's magnitude, and the way the arrow points shows its direction.

In a coordinate system, we can describe a vector by its components. For the vector $\vec{v}$ in the diagram above, it moves 3 units along the x-axis and 4 units along the y-axis. We can write this in a couple of ways:

• Column notation: $\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$
• Coordinate notation: $\vec{v} = (3, 4)$

This works in three dimensions, too. A vector moving 2 units along the x-axis, 5 units along the y-axis, and -1 unit along the z-axis would be written as $\begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix}$.

When are two vectors considered equal? It’s simple: they must have the same magnitude and the same direction. Where a vector starts doesn't matter for equality. A vector from (0,0) to (2,3) is equal to a vector from (5,5) to (7,8) because both represent a movement of 2 units right and 3 units up.

Now let's look at a couple of special vectors that are fundamental to working with them.

In 2D, the zero vector is $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$. In 3D, it's $\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$. It represents no change in position.

Another crucial concept is the unit vector. A unit vector has a magnitude of exactly 1. Its only purpose is to point in a direction. Think of it as a pure direction, with all the magnitude stripped away.

Any vector (except the zero vector) can have its own corresponding unit vector. To find it, you divide the vector by its own magnitude. This scales the vector down to a length of 1 while preserving its direction. For example, the vector $\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ has a magnitude of $\sqrt{3^2 + 4^2} = 5$. Its unit vector is:

Unit vectors are the basic building blocks for describing direction in space, and they are incredibly important in physics and engineering.

Time to check your understanding of these core concepts.

With these basics down, you're ready to explore how we can combine and manipulate vectors.