Cosine Similarity Calculator

Cosine similarity calculates the cosine of the angle between two non-zero vectors in an inner product space. Mathematically, it is defined as the dot product of vectors A and B divided by the product of their Euclidean magnitudes: Similarity = (A • B) / (||A|| × ||B||). The resulting value ranges from -1 to 1. A score of 1 indicates vectors pointing in the exact same direction (perfect similarity), 0 indicates orthogonality (independence), and -1 indicates exact opposition.

Consider two vectors representing word frequencies in two short documents: Vector A = [3, 0, 1] and Vector B = [2, 1, 0]. First, compute the dot product: (3×2) + (0×1) + (1×0) = 6. Next, calculate the magnitude of Vector A: sqrt(3^2 + 0^2 + 1^2) = sqrt(10) ≈ 3.162. Then, calculate the magnitude of Vector B: sqrt(2^2 + 1^2 + 0^2) = sqrt(5) ≈ 2.236. Finally, divide the dot product by the product of the magnitudes: 6 / (3.162 × 2.236) = 6 / 7.07 = 0.849. This high positive score shows strong thematic alignment between the documents.

Always normalize your data before computation if your variables operate on vastly different scales. Keep in mind that cosine similarity focuses strictly on orientation rather than magnitude, making it ideal for sparse data like text mining where document length can vary wildly. Avoid using this metric when absolute quantities carry crucial analytical weight.