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
Dot Product Result
Vector A:
Vector B:
Dot Product (A·B):
(0, 0)
(0, 0)
0
Formula Used: a₁b₁ + a₂b₂
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.