Dot Product CalculatorFree online math calculator — instant results, no sign-up
Calculate the dot product of two three-dimensional vectors.
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About the Dot Product Calculator
The Dot Product Calculator calculates the scalar product of two three-dimensional vectors from their x, y, and z components. It multiplies corresponding components and adds the three products, giving 32 for the default vectors (1, 2, 3) and (4, 5, 6).
- First x
- The x-component of the first vector, defaulting to 1, is unitless for an abstract vector or expressed in the first vector's physical units.
- First y
- The y-component of the first vector, defaulting to 2, uses the same units as its x- and z-components.
- First z
- The z-component of the first vector, defaulting to 3, uses the same units as its x- and y-components.
- Second x
- The x-component of the second vector, defaulting to 4, is unitless for an abstract vector or expressed in the second vector's physical units.
- Second y
- The y-component of the second vector, defaulting to 5, uses the same units as its x- and z-components.
- Second z
- The z-component of the second vector, defaulting to 6, uses the same units as its x- and y-components.
Formula and method
For a = (a_x, a_y, a_z) and b = (b_x, b_y, b_z), D = a · b = a_x b_x + a_y b_y + a_z b_z. With the default inputs: D = (1 × 4) + (2 × 5) + (3 × 6) = 4 + 10 + 18 = 32.
- aThe first three-dimensional vector.
- bThe second three-dimensional vector.
- a_x, a_y, a_zThe components entered as First x, First y, and First z, respectively.
- b_x, b_y, b_zThe components entered as Second x, Second y, and Second z, respectively.
- ·The dot product operation, which produces a scalar rather than a vector.
- DThe resulting dot product, displayed as Result.
Multiply the two x-components, the two y-components, and the two z-components, preserving any negative signs. Add these three products without taking magnitudes or normalizing either vector. For the default inputs, the component products are 4, 10, and 18, and their sum is 32.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First x1
- First y2
- First z3
- Second x4
- Second y5
- Second z6
Results
- Result32
How to use the Dot Product Calculator
- 1Enter the first vector's components in First x, First y, and First z; the defaults are 1, 2, and 3.
- 2Enter the second vector's components in Second x, Second y, and Second z; the defaults are 4, 5, and 6.
- 3Check that both vectors use the same coordinate axes and that units are consistent within each vector.
- 4Calculate the dot product by multiplying corresponding components and adding the products.
- 5Read the Result row; with the default inputs, it should show 32.
Reading your result
The Result row gives a single scalar, not a vector or an angle. For the defaults, Result = 32. For two nonzero vectors, a positive result indicates an acute angle between them, a negative result indicates an obtuse angle, and zero indicates perpendicular vectors. The result is unitless when both vectors are unitless; otherwise, its units are the product of the two vectors' units.
Limitations
This calculation assumes real-valued, three-dimensional vectors expressed in the same orthonormal Cartesian coordinate system. It does not calculate a cross product, normalize vectors, or directly return the angle between them. A zero vector produces a zero dot product with every vector, but has no defined direction or angle. Physical units must be handled consistently by the user, and decimal or very large inputs may be subject to numerical rounding.
Common questions
Why do the default vectors produce 32?+
The calculator pairs matching coordinates: 1 × 4 = 4, 2 × 5 = 10, and 3 × 6 = 18. Adding these products gives 32.
Does swapping the first and second vectors change the dot product?+
No. The real-valued dot product is commutative, so (1, 2, 3) · (4, 5, 6) and (4, 5, 6) · (1, 2, 3) both equal 32.
Does a Result of zero always mean the vectors are perpendicular?+
It means the vectors are perpendicular if both are nonzero. If either vector is the zero vector, the result is also zero, but an angle involving that vector is undefined.
Can I use this calculator for two-dimensional vectors?+
Yes. Enter the two x- and y-components and set both First z and Second z to zero. The z-component product then contributes nothing to the sum.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.