Magnitude and Direction

The components of a vector are the two shorter sides of a right-angled triangle whose hypotenuse is the vector itself. So the magnitude — written $|\mathbf{a}|$ or just $a$ — comes straight from Pythagoras.

Magnitude is a length, so it is never negative, and the signs of the components disappear when they are squared. $3\mathbf{i}-4\mathbf{j}$ and $-3\mathbf{i}+4\mathbf{j}$ both have magnitude 5.

The angle a vector makes with the positive $x$-axis satisfies $\tan \theta =\frac{a_2}{a_1}$. But $\arctan$ on a calculator only ever returns an angle between $-90^{\circ }$ and $90^{\circ }$, so it cannot distinguish a vector pointing up-and-right from one pointing down-and-left — the ratio is the same for both.

The fix is to sketch the vector first. Decide from the signs of the components which quadrant it lies in, then adjust the calculator value to match. A quick sketch costs seconds and prevents an answer that is $180^{\circ }$ wrong.

Going the other way is resolving. If a vector has magnitude $r$ and makes angle $\theta$ with the positive $x$-axis, its components are the two sides of the same right-angled triangle: $r\cos \theta$ across and $r\sin \theta$ up.

This is the single most-used step in Mechanics. Every time a force acts at an angle, this is how it is split into horizontal and vertical parts so that the components can be added.

A $\text{unit vector}$ has magnitude 1. It carries direction and nothing else, which makes it the natural way to say "this way, but scaled however you like".

To find the unit vector in the direction of $\mathbf{a}$, divide $\mathbf{a}$ by its own magnitude. The notation $\hat{\mathbf{a}}$ ("a-hat") is standard. It follows that any vector can be written as its magnitude times its unit vector, $\mathbf{a}=|\mathbf{a}|\hat{\mathbf{a}}$ — which is exactly the "size and direction" split made explicit.