Magnitude Formula
The magnitude formula depends on what you're measuring: for a vector v = ⟨ a, b ⟩, magnitude is |v| = √(a^2 + b^2); for a single number, magnitude is the absolute value |x| — its distance from zero.
The Formula
When to use: How big something is, regardless of which way it points—5 miles east and 5 miles west are the same distance.
Quick Example
Notation
What This Formula Means
Magnitude measures the size or length of a quantity — for a vector (a, b), it is sqrt(a^2 + b^2). For a single number, magnitude is its absolute value: how far it is from zero, ignoring sign or direction.
How big something is, regardless of which way it points—5 miles east and 5 miles west are the same distance.
Formal View
Worked Examples
Example 1
easyAnswer
First step
Full solution
- 2 because is already 7 units from zero (positive, so unchanged).
- 3 because 0 is 0 units from zero.
Example 2
mediumExample 3
mediumCommon Mistakes
- Reporting a magnitude as negative - size and distance are never negative; drop the sign.
- Forgetting magnitude discards direction - 5 east and 5 west have the same magnitude.
- Adding vector components instead of using - magnitude of is the Pythagorean length, not .
Why This Formula Matters
Magnitude separates 'how much' from 'which way', a split that runs through physics (speed vs velocity), distance, and error. It is also always non-negative, which is the key fact that distinguishes it from a signed coordinate. Recognizing it by "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" — rather than by familiar numbers — is what lets a student tell it apart from absolute value and signed integer / coordinate and vector itself in a mixed problem set.
Frequently Asked Questions
What is the Magnitude formula?
Magnitude measures the size or length of a quantity — for a vector (a, b), it is sqrt(a^2 + b^2). For a single number, magnitude is its absolute value: how far it is from zero, ignoring sign or direction.
How do you use the Magnitude formula?
How big something is, regardless of which way it points—5 miles east and 5 miles west are the same distance.
What do the symbols mean in the Magnitude formula?
denotes the magnitude (absolute value) of
Why is the Magnitude formula important in Math?
Magnitude separates 'how much' from 'which way', a split that runs through physics (speed vs velocity), distance, and error. It is also always non-negative, which is the key fact that distinguishes it from a signed coordinate. Recognizing it by "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" — rather than by familiar numbers — is what lets a student tell it apart from absolute value and signed integer / coordinate and vector itself in a mixed problem set.
What do students get wrong about Magnitude?
The procedure for magnitude is the easy part; the trap is reporting a magnitude as negative. Asking "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
What should I learn before the Magnitude formula?
Before studying the Magnitude formula, you should understand: more less, integers.