Vector Magnitude
Length of a vector.
Formula
Variables
Example
|(3,4)|=5
How it works
The magnitude of a vector tells you how long it is, regardless of direction. It comes from the Pythagorean theorem: each component is a perpendicular side, so combining their squares gives the squared length, and taking the square root gives the actual length. In more dimensions, the same idea extends by adding the squares of all components.
When to use it
Use this when you need the size of a vector in geometry, physics, or data analysis. It is especially useful for finding speed from velocity components, distance from an origin, or the length of an arrow in coordinate space.
Worked example, step by step
Find the magnitude of v = (1, 2, 2) Step 1: Square each component: 1^2 = 1, 2^2 = 4, 2^2 = 4 Step 2: Add the squares: 1 + 4 + 4 = 9 Step 3: Take the square root: ||v|| = sqrt(9) Step 4: Simplify: ||v|| = 3
Common mistakes to avoid
- Forgetting to square every component before adding
- Leaving the answer as a squared value instead of taking the square root
- Mixing up vector magnitude with the sum of the components
Did You Know?
A vector's magnitude is just the Pythagorean theorem extended to any number of dimensions.
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