This lesson explains what the magnitude of a vector represents, how to calculate it in two and three dimensions using the Pythagorean-based distance formula, and how to find it for a vector drawn between two points, with fully worked numerical examples.
What Is the Magnitude of a Vector?
A vector has both a size and a direction. The magnitude of a vector is the number that describes that size, in other words, how long the vector is. If you picture a vector as an arrow drawn on a coordinate grid, the magnitude is just the length of that arrow, regardless of which way it points.
Because magnitude measures a length, it is always zero or positive, it is never negative. The magnitude of a vector \(\vec{v}\) is written \(|\vec{v}|\) or sometimes \(\|\vec{v}\|\).
Magnitude Formula for a 2D Vector
If a vector is written in component form as \(\vec{v} = (x, y)\), its horizontal and vertical components form the two legs of a right triangle, and the vector itself is the hypotenuse. That means the magnitude comes straight from the Pythagorean theorem:
\( |\vec{v}| = \sqrt{x^2 + y^2} \)
A vector drawn as the hypotenuse of a right triangle formed by its horizontal and vertical components.
This is the same idea used when you measure horizontal and vertical distances between two points on a grid, since a vector's components are just those horizontal and vertical distances.
Sometimes a vector is described by its starting point and ending point rather than by components right away. If a vector goes from point \(A(x_1, y_1)\) to point \(B(x_2, y_2)\), first find the component form by subtracting coordinates:
\( \vec{AB} = (x_2 - x_1, \; y_2 - y_1) \)
Then apply the usual magnitude formula to that result.
Example 4
Find the magnitude of the vector from \(A(1, 2)\) to \(B(4, 6)\).
This is exactly the distance formula in disguise: the magnitude of a vector between two points is just the distance between those two points.
Magnitude, Unit Vectors, and Other Applications
Once you know a vector's magnitude, you can rescale the vector to have a length of exactly \(1\) by dividing every component by the magnitude. This produces a unit vector pointing in the same direction as the original. Magnitude also plays a key role when describing a vector by its length and its direction angle instead of by \(x\) and \(y\) components.
In physics-style problems, several forces acting on an object combine into a single resultant vector, and the magnitude of that resultant force is found using the same square-root formula applied to the combined components. The same square-root idea also underlies the magnitude of a cross product between two vectors, though that calculation involves an extra step beyond simple addition of components.