What is the Dot Product?
The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This number provides critical information about the relationship between the two vectors, including their relative orientation and magnitude.
In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used in physics, engineering, and computer graphics to calculate work, projections, and angles between forces or objects.
How to Use This Dot Product Calculator
Using our dot product calculator with steps is simple and efficient. Follow these instructions to get your result:
- Input Vector A: Enter the components of your first vector separated by commas. For example, for a 3D vector, you might enter "3, -2, 5".
- Input Vector B: Enter the components of your second vector. Note: Both vectors must have the same number of dimensions (length).
- Click Calculate: Press the button to see the step-by-step breakdown of the multiplication and addition process.
The Mathematical Formula
The algebraic formula for the dot product of two vectors A = [a₁, a₂, ..., aₙ] and B = [b₁, b₂, ..., bₙ] is defined as:
A · B = ∑ (aᵢ * bᵢ) = a₁b₁ + a₂b₂ + ... + aₙbₙ
Frequently Asked Questions
What does a dot product of zero mean? If the dot product of two non-zero vectors is zero, it means the vectors are orthogonal (perpendicular) to each other at a 90-degree angle.
Is the result a vector? No. Unlike the cross product, the dot product results in a scalar (a single real number), not a vector.
Can I use this for 2D and 3D vectors? Yes, this calculator supports any dimension as long as both inputs have the same number of elements.