Vector notation — the language of vectors

A vector is a quantity with a size and a direction, like "5 km north-east" or "a force of 20 N straight down". A plain number with no direction, like a mass of 3 kg, is called a scalar.

In this topic a vector is usually a move (a displacement) from one place to another. Picture it as an arrow: its length is the size of the move, and it points the way you go. An arrow can be drawn anywhere; only its length and direction matter.

To describe a move in 3D, say how far it goes in each of the three axis directions: $x$ (across), $y$ (back or forward) and $z$ (up). Those three numbers are the vector's components.

Take the move "3 along $x$, 2 along $y$, 4 up $z$". Cambridge writes it in any of these ways:

1. Column vector: the components stacked in a bracket, $\begin{pmatrix} 3 \\ 2 \\ 4 \end{pmatrix}$. Top is $x$, middle is $y$, bottom is $z$. (In 2D there are just two rows, $x$ then $y$.)
2. $\mathbf{i}, \mathbf{j}, \mathbf{k}$ form: $3\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$. Here $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are arrows of length 1 pointing along the $x$-, $y$- and $z$-axes. So $3\mathbf{i}$ means "3 steps along $x$", and so on. (Arrows of length 1 are called unit vectors.)
3. Bold letter: give the vector a name, like $\mathbf{a}$. In handwriting you can't do bold, so underline it: $\underline{a}$.
4. Arrow over two letters: $\overrightarrow{OA}$ means "the move from point $O$ to point $A$".

They're four names for one thing, so you can swap between them freely: just read off the three numbers.