Vector Calculator
Enter two three-component vectors to calculate their sum, difference, dot product, cross product, magnitudes and angle, with an XY-projection diagram.
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Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Add or subtract corresponding components. Sum componentwise products for the dot product. Use the determinant component formula for A×B. The angle uses atan2 of normalized cross-product magnitude and dot product, which is stable near parallel directions.
A·B = AₓBₓ+AᵧBᵧ+A_zB_z; A×B = (AᵧB_z−A_zBᵧ, A_zBₓ−AₓB_z, AₓBᵧ−AᵧBₓ)
Worked example
For A=(1,2,3) and B=(4,5,6), A·B=32 and A×B=(−3,6,−3). Their magnitudes are √14 and √77, and their angle is about 12.9331545°.
Assumptions and input details
Use one consistent coordinate basis and compatible component units. The angle is the smaller 3D angle from 0° to 180° and is undefined if either vector is zero. The drawing is only the XY projection: it excludes z and must not be read as the 3D angle. A×B changes sign when the order is reversed. Products use floating-point arithmetic; very small components may underflow, so rescale extreme inputs when needed.
References: OpenStax: vectors
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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