Dot Product Calculator
The dot product combines corresponding components of two three-dimensional vectors. It is used in projections, angle calculations, work, and other vector relationships.
How to use this calculator
- Enter x, y and z components for each vector. The calculator shows the scalar product, both magnitudes and the angle when defined.
- Check the labels and units before applying the result to a real task.
- Change one input at a time when comparing scenarios.
Formula and method
A·B = aₓbₓ + aᵧbᵧ + a_zb_z = |A||B|cos(θ)
- A·B = aₓbₓ + aᵧbᵧ + a_zb_z = |A||B|cos(θ)
- Vector components must use compatible units and coordinate conventions. The angle is undefined if either vector has zero magnitude.
Worked examples
Worked example
For A = (1, 2, 3) and B = (4, 5, 6), calculate the scalar product.
A·B = 1×4 + 2×5 + 3×6 = 32.
A·B = 1×4 + 2×5 + 3×6 = 32.
Understanding the result
The dot product combines corresponding components of two three-dimensional vectors. It is used in projections, angle calculations, work, and other vector relationships.
Vector components must use compatible units and coordinate conventions. The angle is undefined if either vector has zero magnitude.
Important notes
Vector components must use compatible units and coordinate conventions. The angle is undefined if either vector has zero magnitude.
Displayed values are rounded for readability; retain appropriate precision when continuing a calculation.
Frequently asked questions
Is the dot product a vector?
No. The dot product returns a scalar number. The cross product, by contrast, returns a vector in three dimensions.
What does a zero dot product mean?
For non-zero vectors, a zero dot product means the vectors are perpendicular. A zero vector also makes the dot product zero, so check magnitudes.
Can vector units be mixed?
Components should be expressed consistently. If A and B represent physical quantities, the resulting units are the product of their component units.
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This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.