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Vector Angle Calculator

Compare the directions of two vectors without confusing direction with length. Enter Cartesian components for both vectors using the same axes. The calculator returns the smaller unsigned angle from 0° through 180°, together with the equivalent radians. This is useful for checking alignment, perpendicular directions and the angle in a dot-product problem. Scaling either vector by a positive number leaves its direction, and therefore this angle, unchanged.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

Angle in degrees90
Angle in radians1.570796327

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

θ = arccos[(A · B)/(|A||B|)].

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

For A = (1, 0, 0) and B = (0, 1, 0), the dot product is zero. arccos(0) = 90° = π/2 radians.

Limitations and interpretation

The angle is undefined if either vector is zero. No clockwise or counterclockwise orientation is inferred from this unsigned angle. Negative scaling reverses a direction. Values very near parallel or antiparallel directions may lose angular precision in floating-point arithmetic.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

Can it tell me clockwise versus counterclockwise?

No. This is an unsigned spatial angle. A signed planar angle requires an orientation convention and different inputs.

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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