How do you find the angle between two vectors?

Use θ = arccos((A·B) / (|A||B|)) when both vectors have non-zero magnitude.

Can this calculator work for 2D vectors?

Yes. For 2D vectors, just set the z-components to zero.

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Dot Product Calculator

Q: What is the dot product of two vectors?

The dot product of vectors A and B is A·B = AxBx + AyBy + AzBz. It is also equal to |A||B|cosθ.

Q: Is dot product the same as scalar product?

Yes. Dot product and scalar product refer to the same vector operation.

Use this dot product calculator to calculate the vector dot product, also called the scalar product or inner product. The calculator can find the dot product from vector components or from magnitudes and angle, and it also shows the angle between two vectors.

Vector Dot Product, Scalar Product and Angle Calculator

Change any input and the calculator updates automatically without pressing Calculate.

Calculator Inputs

Input mode

Vector Components

Use this as a 2D or 3D vector dot product calculator. For 2D vectors, set z = 0.

Vector A = (Ax, Ay, Az)

Vector B = (Bx, By, Bz)

Magnitudes and Angle

Use this as a dot product calculator with angle: A·B = |A||B|cosθ.

|A|

|B|

θ (deg)

Use dot “.” as decimal separator.

Results

A · B

Dot Product

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Magnitude of A

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Magnitude of B

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Angle Between Vectors

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Interpretation

Enter values to compute the dot product and the angle between vectors.

Live Vector Diagram

Step-by-step result

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For true 3D vectors, the calculator still computes the correct dot product and angle. The live diagram is shown only for 2D visualization.

Dot Product Formula

A · B = AxBx + AyBy + AzBz

A · B = |A||B|cosθ

θ = arccos((A · B) / (|A||B|))

The dot product can be found directly from vector components or from magnitudes and the angle between the vectors. This is why the page also works as an angle between two vectors calculator and a dot product calculator with angle.

What the Dot Product Means

The dot product measures how much one vector points in the same direction as another. A positive result means the vectors point mostly in the same direction, a negative result means they point mostly in opposite directions, and a zero result means they are perpendicular.

How to Use This Scalar Product Calculator

Select Vector Components or Magnitudes and Angle.
Enter the vector values.
The calculator updates instantly when you change any value.
Read the dot product, vector magnitudes, and the angle between the vectors.

Worked Example

Let A = (1, 2, 0) and B = (3, 4, 0).

A · B = 1×3 + 2×4 + 0×0 = 11
|A| = √(1² + 2²) = √5 ≈ 2.236
|B| = √(3² + 4²) = 5
θ = arccos(11 / (2.236 × 5)) ≈ 10.3°

Frequently Asked Questions

What is the dot product of two vectors?

The dot product is a scalar value found from the sum of the products of corresponding vector components.

Is dot product the same as scalar product?

Yes. Dot product, scalar product, and inner product are often used to describe the same operation in this context.

What does a zero dot product mean?

A zero dot product means the vectors are perpendicular.

Can this calculator find the angle between vectors?

Yes. When both vectors have non-zero magnitude, it calculates the angle using the inverse cosine formula.

Related Calculators

Physics Calculators 2D Vector Addition Calculator Vector Addition Calculator Vector Cross Product Calculator Vector Magnitude and Angle Calculator

References

Giancoli, Douglas C. Physics: Principles with Applications. Pearson.
Halliday, Resnick, Walker. Fundamentals of Physics.
OpenStax. University Physics. Vector algebra and dot product sections.