Dot products and cosine similarity

Goal of this lesson

Calculate dot products and cosine similarity by hand, explain why magnitude changes a dot-product ranking, and draw a clear boundary between normalization, a model’s training convention, and the product goal.

Dot product

$x\cdot y={\sum }_i{x}_i{y}_i$

It can also be written as:

$x\cdot y=\Vert x\Vert \Vert y\Vert \cos \theta$

It depends on both direction and the lengths of the two vectors. $(1,2)\cdot (3,4)=11$. A zero dot product means the vectors are orthogonal in Euclidean geometry; a positive or negative result means their angle is less than or greater than 90°, respectively.

Cosine similarity

$\cos (x,y)=\frac{x\cdot y}{\Vert x{\Vert }_2\Vert y{\Vert }_2}$

For nonzero vectors, the value lies in $[-1,1]$. It compares direction and ignores uniform scaling: (1,2) and (10,20) have cosine similarity 1.

If every vector is L2-normalized:

$\cos (\hat{x},\hat{y})=\hat{x}\cdot \hat{y}$

Some systems can therefore use an inner-product index to implement cosine similarity, but only when both query and document vectors are normalized in the same way.

A worked comparison

For query $q=(1,1)$:

• $a=(2,2)$: same direction; cosine = 1 and dot product = 4.
• $b=(10,0)$: cosine = $1/\sqrt{2}\approx 0.707$ and dot product = 10.

Dot product places the much larger-magnitude vector $b$ first; cosine places the directionally identical vector $a$ first. Which is correct depends on the embedding’s training objective and the product semantics.

Uses in retrieval

• Dual-encoder models often use dot product as their training and retrieval score; vector magnitude may carry information.
• General text embeddings often recommend cosine or an equivalent normalized dot product, but you must check the model card or official documentation.
• Similarity is a candidate-ranking signal, not proof of factual correctness, answerability, or authorization.
• A high-score threshold cannot be copied across corpora, models, and query types. Calibrate it on a representative labeled set.

Common misconceptions

• Explaining 0.8 as an “80% probability of relevance.” Cosine is not a probability.
• Assuming negative cosine always means “irrelevant”; validate the distribution and task for the model in use.
• Normalizing the query but not documents, then claiming to use cosine.
• Attributing a large dot product to directional similarity while ignoring magnitude.

Exercises and self-check

1. Calculate the dot product and cosine for $q=(1,0)$ with (2,0), (1,1), and (-1,0).
2. Prove that squared Euclidean distance between unit vectors is $\Vert x-y{\Vert }^2=2-2x\cdot y$, then explain how cosine ranking and Euclidean-distance ranking relate in this case.

• I can explain how dot product and cosine differ with respect to magnitude.
• I do not treat cosine as a probability.
• I know the metric must match the model’s training convention.

References

• Google: Measuring Similarity from Embeddings
• scikit-learn 1.9.0: Cosine similarity

Sources verified on 2026-07-14. Next: Distance, normalization, and high-dimensional spaces.