Dot Product Calculator

Compute the scalar product of two vectors in 2D or 3D space

Mathematical Note: The dot product is a scalar value that measures the parallel components of two vectors multiplied together.

Tip: Enter vector components as numbers (integers or decimals). Negative values are allowed.

Vector Dimension

Vector A

Vector B

Angle Between Vectors (Optional)

°

If provided, the calculator will verify the result using A·B = |A||B|cosθ

Understanding the Dot Product

Algebraic Definition

  • For 2D vectors: A·B = a₁b₁ + a₂b₂
  • For 3D vectors: A·B = a₁b₁ + a₂b₂ + a₃b₃
  • Sum of corresponding component products
  • Results in a scalar (single number)

Geometric Interpretation

  • A·B = |A||B|cosθ (θ = angle between vectors)
  • Measures how much vectors point in same direction
  • Positive when θ < 90°, negative when θ > 90°
  • Zero when vectors are perpendicular (θ = 90°)

Practical Applications

Physics: Calculating work done (W = F·d where F is force and d is displacement).

Computer Graphics: Determining lighting (dot product between surface normal and light direction).

Machine Learning: Measuring similarity between feature vectors (cosine similarity uses normalized dot product).

Frequently Asked Questions

What's the difference between dot product and cross product?

The dot product results in a scalar and measures parallel components, while the cross product results in a vector and measures perpendicular components. Dot product: A·B = |A||B|cosθ. Cross product: |A×B| = |A||B|sinθ.

Can dot product be negative?

Yes, the dot product can be negative when the angle between vectors is greater than 90°. This indicates the vectors point in generally opposite directions.

What does a dot product of zero mean?

A zero dot product means the vectors are perpendicular (orthogonal). The angle between them is exactly 90°, so cos(90°) = 0.

How is dot product related to vector length?

The dot product of a vector with itself (A·A) equals the square of its length (|A|²). So vector length can be found as √(A·A).

Why is the angle input optional?

The dot product can be calculated two ways: from components (a₁b₁ + a₂b₂...) or from magnitudes and angle (|A||B|cosθ). We include the angle option to let you verify calculations using both methods.

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