HesapHanem

3D Vector Calculator

Find magnitude and unit vector of a 3D vector

How it is calculated

The magnitude is the length of the vector: |v| = √(x² + y² + z²), where x, y and z are the components along the three axes in the same unit (metres, newtons, any unit). The unit vector divides each component by that length: û = (x/|v|, y/|v|, z/|v|). The unit vector has length 1 and no unit of its own. Results are rounded to two decimals for display, so the rounded unit components may not square-sum to exactly 1. The zero vector (0, 0, 0) has no direction, so the tool returns no result for it. Negative components are fine and flip the sign of that component.

Worked example

Take v = (3, 4, 12). Squares: 9 + 16 + 144 = 169. Magnitude: √169 = 13.00. Unit vector: 3/13 = 0.23, 4/13 = 0.31, 12/13 = 0.92. So the direction of v is (0.23, 0.31, 0.92).

Try this example →

Common mistakes

  • Entering a 2D vector and leaving z blank. All three components are required, so type 0 for z.
  • Mixing units between components, for example x in metres and y in centimetres. The magnitude is meaningless unless all three share one unit.
  • Expecting the unit vector to keep the original length. It always has length 1; only the direction is preserved.
  • Trying to normalise (0, 0, 0). A zero vector has no direction, so there is no result.

About this calculator

From a vector's components (x, y, z), the calculator finds the magnitude with the 3D distance formula, |v| = √(x² + y² + z²), then divides each component by that magnitude to produce the unit vector — a vector of length 1 pointing in the exact same direction, the form most physics and graphics formulas require.

Physics students use it to resolve force, velocity, and displacement vectors in three dimensions, and game developers and 3D graphics programmers use unit vectors constantly for lighting normals, camera directions, and movement — anywhere a direction needs to be represented independently of its original length.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Unit vectors earn their keep in graphics and game code. A diagonal movement of (1, 1, 0) has length 1.41, so skipping normalisation makes a character about 41 percent faster on the diagonal than along an axis. The tool shows components to two decimals only, so for code compute from the original components instead of copying the rounded values. Is your vector a force, a velocity or a direction in a 3D scene?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators