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Math

3D Vector Calculator

Find vector sums, differences, dot and cross products, magnitudes and the angle between two vectors.

Exact method explained belowEditable inputs
Calculation notebook
Calculation notebook

The formula and worked example below explain the calculation.

See the method

Enter your values

Runs on your device

Your answer

Your result

Let’s calculate.

Use the Calculate button to see your result.

Review the method below for assumptions and conventions.

How to use this tool

  1. Enter vector a: x, y, z, vector b: x, y, z.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

The method, explained

Dot product is Σaᵢbᵢ. Cross product uses component determinants. Magnitude is sqrt(Σaᵢ²); the angle is acos(dot/(|a||b|)) for nonzero vectors.

A WORKED EXAMPLE

Using vector a: x, y, z = 1, 0, 0, vector b: x, y, z = 0, 1, 0, the result is 0. Change these example inputs to match your task; use the method above to check each step.

Understanding your result

A zero vector has no direction. Its dot and cross products still exist, but the angle formula would divide by its zero magnitude.

What to keep in mind

Cartesian components in an orthonormal right-handed coordinate system. The main result is the dot product. Finite-precision arithmetic is used. Check the domain and the stated convention before using an approximation.

Common questions

Why is the angle undefined for a zero vector?

A zero vector has no direction. Its dot and cross products still exist, but the angle formula would divide by its zero magnitude.

How can I check the result?

Dot product is Σaᵢbᵢ. Cross product uses component determinants. Magnitude is sqrt(Σaᵢ²); the angle is acos(dot/(|a||b|)) for nonzero vectors. The worked example uses editable inputs. Calculations run on your device; no external API or account is required.

Methodology maintained by ClarityKit. How these tools are built and checked.