In this explainer, we will learn how to determine the magnitude of two-dimensional vectors.

Recall that a vector has two aspects: direction and magnitude. The magnitude is the vector’s “size ” or “length. ”

### Example 1: Visually Comparing the Magnitude of Vectors

Which vector has the greatest magnitude?

### Answer

Visual inspection tells us that is the longest, so it has the greatest magnitude. Note that we can move to have the same initial point as and then rotate it so that the grid it lies in matches the grid of . This makes its greater length clear.

We denote the magnitude of the vector by and use Pythagoras’ theorem to compute it, against the grid of unit squares.

### Example 2: Finding the Magnitude of a Vector on a Coordinate Grid

The vector is shown on the grid of unit squares below. Find the value of .

### Answer

The magnitude (or length) of the vector is

The magnitude is calculated in the same way from its components.

### Example 3: Finding the Magnitude of a Vector from Its Initial Point and Endpoint

What is the magnitude of the vector , where and ?

### Answer

We have that