Calculate Vector Dot Products Instantly
Calculate the dot product (scalar product) of two vectors quickly and accurately. Our calculator handles 2D and 3D vectors, showing both the calculation process and the geometric interpretation.
Welcome to our Dot Product Calculator! This tool helps you calculate the dot product of two 3D vectors. The dot product is a crucial concept in linear algebra and has various applications in physics and computer graphics.
The dot product of two vectors is a scalar value that gives information about the angle between the vectors and their magnitudes.
For two vectors A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂), the dot product is calculated as:
A · B = x₁x₂ + y₁y₂ + z₁z₂
The dot product is positive if the vectors point in similar directions, negative if they point in opposite directions, and zero if they are perpendicular.
The dot product is a fundamental vector operation that takes two vectors and returns a scalar value. For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the dot product is: a · b = a₁b₁ + a₂b₂ + a₃b₃. The result tells us about the angle and relationship between the vectors.
The dot product relates to the angle θ between vectors: a · b = |a| |b| cos(θ). If the dot product is positive, vectors point in similar directions; if negative, they point in opposite directions; if zero, they are perpendicular (orthogonal).
Dot products are used in: physics for calculating work (force · displacement), computer graphics for lighting calculations, machine learning for similarity measures, geometry for finding angles between vectors, and engineering for projecting forces.
