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Vector Calculator: Magnitude, Dot Product, Cross Product and Angle

This free vector calculator works with vectors of two or more dimensions. It gives the magnitude, unit vector, dot product, cross product (3D), the angle between two vectors, their sum and difference, and the projection.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Vectors describe direction and size. Students, game developers and engineers use these formulas every day.

Enter vector A and, for two-vector results, vector B.

How to Use This Tool

  1. Enter or paste your input in the box or boxes above.
  2. Choose the options, if there are any.
  3. Press the button to run the tool.
  4. Read the result and use the copy button where one is shown.

How the Calculation Works

The magnitude is the square root of the sum of squares. The unit vector divides each part by the magnitude. The dot product multiplies matching parts and adds them. The 3D cross product uses the determinant rule. The angle uses arccos of the dot product over the product of magnitudes. The projection of A on B is (A.B / B.B) x B.

Worked Example

For A = (1, 2, 3) and B = (4, 5, 6): A.B = 32, A x B = (-3, 6, -3), |A| = 3.7417 and the angle is 12.933 degrees.

Common Uses

  • Physics and engineering.
  • Game and graphics code.
  • Maths homework.
  • Checking direction and distance.

Tips and Common Mistakes

  • Use the same number of parts for both vectors.
  • A zero vector has no unit vector or angle.
  • The angle is shown in degrees and radians.
  • Check the first result against a case you already know, so you can trust the tool for the rest.
  • Nothing you type leaves your device. The work is done by your browser, so it also works with a slow connection once the page has loaded.

Next, you may need the matrix calculator, trigonometry calculator, prime number checker, GCD and LCM calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

The sum of the products of matching parts. For (1,2,3) and (4,5,6) it is 1x4 + 2x5 + 3x6 = 32.
A vector at right angles to both A and B, defined for 3D vectors. Its length equals the area of the parallelogram they span.
cos of the angle equals the dot product divided by the product of the lengths.
Yes. The cross product then appears as a single number (the z part).

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.