Case 1
- x component
- 3
- y component
- 4
- z component
- 0
5.000000
Open with these valuesVector Magnitude Calculator
Change any value to update the result.
5.000000
Magnitude: 5.000000Examples
Square each component, add the squares, take the square root. The vector (3, 4, 0) has magnitude 5, and (1, 2, 2) has exactly 3. Leave z at 0 for a vector in the plane; the formula then reads √(x² + y²). The result is never negative.
The numbers at a glance
Worked examples
Case 2
- x component
- 1
- y component
- 2
- z component
- 2
3.000000
Open with these valuesCase 3
- x component
- 1.5
- y component
- 2.5
- z component
- 3.5
4.555217
Open with these valuesHow it's calculated
|v| = √(x² + y² + z²)
- StepEnter the three components; a vector in the plane keeps z at 0.
- StepEach one is squared and the three squares are added.
- ResultThe square root of that sum is the length of the vector.5.000000
Reference table
| x, y, z | Exact | Magnitude |
|---|---|---|
| 0, 0, 0 | √0 | 0.000000 |
| 1, 1, 1 | √3 | 1.732051 |
| 1, 2, 2 | √9 | 3.000000 |
| 1.5, 2.5, 3.5 | √20.75 | 4.555217 |
| 3, 4, 0 | √25 | 5.000000 |
| 2, 3, 6 | √49 | 7.000000 |
Questions
How do I calculate the magnitude of a vector?
Square each component, add the squares together, then take the square root: |v| = √(x² + y² + z²). The vector (3, 4, 0) gives √(9 + 16 + 0) = √25 = 5.
What is the magnitude of a vector?
It is the length of the vector — the straight-line distance from the origin to the point the vector reaches. It captures how big the vector is while ignoring its direction, which is why it is also called the norm.
How does this relate to the Pythagorean theorem?
It is the Pythagorean theorem with one more axis. In the plane the length of (x, y) is √(x² + y²), exactly the hypotenuse of a right triangle; a z-axis simply adds another squared term.
Can a vector's magnitude be negative?
No, it is always zero or positive. Each component is squared before being added, so negative components contribute positive amounts. The only vector with magnitude zero is the zero vector (0, 0, 0).
How do I find the magnitude of a 2D vector?
Leave the z component at 0 and the formula reads |v| = √(x² + y²). The 2D vector (6, 8) then has magnitude √(36 + 64) = √100 = 10.