Dot Product (Scalar Product) Calculator
The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single scalar value representing the projection of one vector onto another.
Formula first
Overview
Geometrically, the dot product relates the magnitudes of two vectors and the cosine of the angle between them. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. It is a fundamental operation in vector spaces, serving as the basis for defining orthogonality and vector projections.
Symbols
Variables
a b = Dot Product, = Vector A component 1, = Vector A component 2, = Vector B component 1, = Vector B component 2
Apply it well
When To Use
When to use: Use the dot product when you need to determine the angle between two vectors, check if two vectors are orthogonal (perpendicular), or calculate the work done by a force vector acting over a displacement.
Why it matters: The dot product is essential in physics for energy calculations, in computer graphics for lighting and shading algorithms, and in machine learning for measuring similarity between data points.
Avoid these traps
Common Mistakes
- Confusing the dot product with the cross product, which results in a vector rather than a scalar.
- Forgetting that the result of a dot product is a scalar value, not a vector.
One free problem
Practice Problem
Calculate the dot product of vector a = [3, 2] and vector b = [1, 4].
Solve for: dotProduct
Hint: Multiply the corresponding components (3*1) and (2*4), then add the results together.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
- Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
- Stewart, J. (2015). Calculus: Early Transcendentals, 8th Edition.