Vector Magnitude
Length of a vector.
Formula
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Example
|(3, 4, 0)| = √(9+16+0) = 5.
How it works
The magnitude of a vector is its distance from the origin, so it tells you how long the vector is in space. The formula comes from the Pythagorean theorem: in 2D you find the length of the hypotenuse from the horizontal and vertical components, and in 3D the third component adds another squared term. Squaring each component makes negative directions contribute positively to length, which is why the result is always nonnegative.
When to use it
Use this formula whenever you need the size or length of a vector in geometry, physics, or engineering. Common examples include finding displacement, velocity magnitude, force strength, or checking the distance from the origin in 3D coordinates.
Worked example, step by step
Find the magnitude of v = (2, -5, 4). |v| = √(v1^2 + v2^2 + v3^2) |v| = √(2^2 + (-5)^2 + 4^2) |v| = √(4 + 25 + 16) |v| = √45 |v| = 3√5
Common mistakes to avoid
- Forgetting to square negative components before adding them
- Adding the components first instead of adding their squares
- Leaving the answer as a squared length instead of taking the square root
Did You Know?
A vector's magnitude is the Pythagorean theorem generalized to any number of dimensions.
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