Dot Product Calculator

Vector Projection and Angles

The Dot Product Calculator is an indispensable tool for vector analysis. While the cross product results in a new vector, the dot product results in a scalar (a single number). This calculator computes the dot product, the magnitudes, and the exact angle between the vectors.

Two Ways to Calculate

The beauty of the dot product is that it links pure algebra to pure geometry. There are two definitions that are mathematically equivalent:

1. Algebraic: Multiply matching coordinates and sum them up ($A_x B_x + A_y B_y + A_z B_z$).
2. Geometric: Multiply the magnitudes and the cosine of the angle between them ($|A||B|\cos(\theta)$).

Because these are equal, we can use the simple algebraic method to easily solve for the unknown geometric angle!

Real-World Applications

• Physics (Work): Work is defined as the dot product of Force and Displacement ($W = F \cdot d$). Only the force applied in the direction of movement contributes to work.
• Computer Graphics: Used extensively in lighting shaders (Lambertian reflectance) to determine how directly a light source is hitting a 3D polygon.
• Machine Learning: The core mathematical operation inside artificial neural networks, used to calculate the weighted sum of inputs.