The choice of distance metric directly impacts which vectors are considered “nearest” in a search by emphasizing different geometric or algebraic relationships between vectors. Euclidean distance, cosine similarity, and dot product each prioritize distinct aspects of vector relationships, such as magnitude, direction, or a combination of both. This leads to variations in neighbor rankings, even for the same dataset.
Euclidean distance measures the straight-line distance between two vectors in space. It treats vectors as points in a geometric space and is sensitive to both the direction and magnitude of the vectors. For example, two vectors with small coordinate-wise differences will have a low Euclidean distance, even if their magnitudes differ. However, if one vector is significantly longer (e.g., [3, 4] vs. [1, 1]), their Euclidean distance may be large despite similar directional alignment. This makes Euclidean distance suitable for applications where absolute position matters, such as physical coordinates or normalized embeddings where magnitude carries meaning. In contrast, cosine similarity focuses on the angle between vectors, ignoring their magnitudes. It ranges from -1 (opposite directions) to 1 (identical directions). For instance, a vector [2, 3] and its scaled version [4, 6] have a cosine similarity of 1, as they point in the same direction. This makes cosine ideal for text embeddings (e.g., TF-IDF or word2vec) where document length or frequency shouldn’t dominate similarity. However, cosine can fail if magnitude differences are meaningful, like in image pixel intensities.
The dot product combines aspects of both magnitude and direction. Mathematically, it equals the product of the vectors’ magnitudes and their cosine similarity. For example, a long vector aligned with a shorter one (e.g., [5, 5] · [2, 2] = 20) may have a higher dot product than two shorter, perfectly aligned vectors (e.g., [1, 1] · [1, 1] = 2). This makes dot product useful when both direction and magnitude matter, such as in recommendation systems where a user’s preference strength (magnitude) and item relevance (direction) are both important. However, dot product can disproportionately favor larger vectors, which may not always be desirable. To mitigate this, vectors are often normalized before applying dot product, effectively converting it into cosine similarity. Choosing between these metrics depends on the problem’s needs: Euclidean for magnitude-sensitive tasks, cosine for direction-focused comparisons, and dot product for hybrid scenarios where magnitude carries weight. Understanding these differences ensures the metric aligns with the data’s inherent properties and the application’s goals.