Vector Magnitude CalculatorFree online math calculator — instant results, no sign-up
Calculate the magnitude of a three-dimensional vector.
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About the Vector Magnitude Calculator
The Vector Magnitude Calculator finds the length of a three-dimensional vector from its x, y, and z components. It uses the three-dimensional Pythagorean theorem to measure the vector's distance from the origin. With the default components x = 3, y = 4, and z = 12, the magnitude is 13.
- x component
- The vector's signed component along the x-axis, defaulting to 3, must use the same units as the y and z components.
- y component
- The vector's signed component along the y-axis, defaulting to 4, must use the same units as the x and z components.
- z component
- The vector's signed component along the z-axis, defaulting to 12, must use the same units as the x and y components.
Formula and method
||v|| = √(x² + y² + z²) = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13
- vThe three-dimensional vector with components (x, y, z).
- xThe signed component along the x-axis.
- yThe signed component along the y-axis.
- zThe signed component along the z-axis.
- ||v||The magnitude, or nonnegative length, of the vector.
Square each component, add the three squares, and take the nonnegative square root of the sum. For the default inputs, the squares are 9, 16, and 144; their sum is 169, whose square root is 13.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- x component3
- y component4
- z component12
Results
- Result13
How to use the Vector Magnitude Calculator
- 1Enter the x component; keep 3 to follow the default example.
- 2Enter the y component; keep 4 to follow the default example.
- 3Enter the z component; keep 12 to follow the default example.
- 4Confirm that all components use the same units and calculate the magnitude.
- 5Read the Result row: the default vector (3, 4, 12) has magnitude 13.
Reading your result
The Result row gives the vector's magnitude, not its direction or individual components. For (3, 4, 12), Result = 13 means the vector's length is 13 in the same units as the inputs. If the components are unitless, the result is also unitless.
Limitations
The formula assumes real-valued components in an orthogonal Cartesian coordinate system with equally scaled axes. All components must represent the same type of quantity and use matching units; convert mixed units before entering them. The result does not specify direction, and different vectors can have the same magnitude. Decimal results may be rounded by the calculator's display.
Common questions
Why does the vector (3, 4, 12) have magnitude 13 rather than the sum of its components?+
The components lie along perpendicular axes, so length is found by adding their squares and taking a square root: √(9 + 16 + 144) = 13. Adding the components directly does not measure the vector's straight-line length.
Can I enter negative x, y, or z components?+
Yes. A negative component indicates direction along its axis. Because each component is squared, changing only its sign does not change the magnitude.
Can this three-dimensional calculator handle a two-dimensional vector?+
Yes. For a vector in the xy-plane, enter 0 for the z component. The formula then reduces to √(x² + y²).
When is the vector magnitude zero?+
The magnitude is zero only when x, y, and z are all zero. Any nonzero real component contributes a positive square, making the magnitude positive.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.