The formula this page uses
|v| = √(v₁² + v₂² + … + vₙ²) distance PQ = |Q − P|
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Magnitude Of A Vector Calculator
Solve magnitude of a vector problems with clear steps, notation, and a final check.
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Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Vectors
Magnitude Of A Vector Calculator explained
The short version
- A vector's magnitude is its length, found by squaring every component, adding, and taking the square root.
- It is never negative, even when every component is negative.
- The same calculation gives the distance between two points, once you subtract to get the vector between them.
What each part means
SymbolWhat it means
v₁, v₂, v₃ — ComponentsThe vector's numbers along each axis, all in the same unit.
|v| — MagnitudeThe length. Same unit as the components, and always zero or positive.
P, Q — Two pointsSubtract P from Q component by component to build the vector between them.
n — Number of componentsThe formula does not change with dimension; you just square more numbers.
Show your work: a full example
- The vectorv = (2, 3, 6)
- Square each component2² = 4, 3² = 9, 6² = 36
- Add the squares4 + 9 + 36 = 49
- Take the square root|v| = √49 = 7
- Use the same steps as a distancefrom P(1, −2, 4) to Q(3, 1, 10), the difference is (3−1, 1−(−2), 10−4) = (2, 3, 6)
- Same vector, so same lengththe two points are 7 units apart
A second, different case
- A different case: negative components and real unitsa plane's velocity is (−120, 160) km/h
- Square each component; the minus disappears(−120)² = 14400 and 160² = 25600
- Add14400 + 25600 = 40000
- Root|v| = √40000 = 200 km/h, the plane's speed
- Note what the sign didthe −120 says it is heading west, but speed itself cannot be negative
- Double every component(−240, 320) gives √(57600 + 102400) = √160000 = 400 km/h
- Read the patternscaling a vector by k scales its magnitude by |k|, so doubling the components doubled the speed
Copy-ready example
v = (2, 3, 6)
The vector
Questions about the Magnitude Of A Vector Calculator
Is magnitude the same as the distance formula?
Yes, they are one formula wearing two names. The distance between P(1, −2, 4) and Q(3, 1, 10) is the magnitude of the vector Q − P = (2, 3, 6), which is why both give 7.
Why does the formula work in 3D and beyond?
Apply Pythagoras twice. In 3D, √(2² + 3²) = √13 is the shadow in the xy-plane, and then √((√13)² + 6²) = √49 = 7 stands that shadow up against the z-component. The same nesting continues into higher dimensions.
What is the magnitude of a velocity vector called?
Speed. Velocity carries direction, speed does not, which is why the plane above has a velocity of (−120, 160) km/h but a speed of 200 km/h. The same split names force versus magnitude of force.
How does scaling a vector change its magnitude?
It multiplies it by the size of the scale factor. Doubling gave 400 km/h above, and multiplying by −1 would leave the magnitude at 200 while reversing the direction entirely.