Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
dimensionless, from −1 to 1; for nonnegative vectors from 0 to 1
| Symbol | Meaning | Unit |
|---|---|---|
| Nonzero vectors of equal dimension | vectors | |
| Vector dot product | product-dependent | |
| ‖x‖2, ‖y‖2 | Euclidean norms of the vectors | same scale as the corresponding vector |
\mathrm{cosSim}(x,y)=\frac{x\cdot y}{\|x\|_2\,\|y\|_2}Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it usually ranges from zero to one; signed features can yield negative values. It is not a general metric distance and should not be confused with Euclidean distance.
Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it usually ranges from zero to one; signed features can yield negative values. It is not a general metric distance and should not be confused with Euclidean distance.
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for Cosine similarity between vectors to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it.
Formula cluster
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
For x=(1,1) and y=(1,0), x·y=1, ‖x‖=√2, ‖y‖=1 and cosine similarity=1/√2≈0.707.
Rank documents, embeddings and feature vectors by directional similarity.
Here is how to work through a typical Cosine similarity between vectors (cosSim = (x·y)/(‖x‖‖y‖)) task step by step:
Task
Use the example for Cosine similarity between vectors to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
The relation applies only under its stated assumptions.
The relation \mathrm{cosSim}(x,y)=\frac{x\cdot y}{\|x\|_2\,\|y\|_2} summarizes Cosine similarity compares the direction of two nonzero vectors independently of their positive scale. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For x=(1,1) and y=(1,0), x·y=1, ‖x‖=√2, ‖y‖=1 and cosine similarity=1/√2≈0.707., symbols are connected to concrete values and the result is interpreted in context. Validity limit: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it Source: scikit-learn User Guide, Pairwise metrics and cosine similarity.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For x=(1,1) and y=(1,0), x·y=1, ‖x‖=√2, ‖y‖=1 and cosine similarity=1/√2≈0.707. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it Source: scikit-learn User Guide, Pairwise metrics and cosine similarity.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it Source: scikit-learn User Guide, Pairwise metrics and cosine similarity.
The stated units are: dimensionless, from −1 to 1; for nonnegative vectors from 0 to 1. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it Source: scikit-learn User Guide, Pairwise metrics and cosine similarity.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn User Guide, Pairwise metrics and cosine similarity. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: Normalizing the dot product removes positive vector length and compares the angle between vectors. The measure is defined only for nonzero vectors. In text retrieval with nonnegative term weights, it Source: scikit-learn User Guide, Pairwise metrics and cosine similarity.
Quanta exam set
Curated exam set for "Cosine similarity between vectors":
Question (front)
Which formula describes Cosine similarity between vectors?
Answer in your set
Question (front)
How do you rearrange Cosine similarity between vectors for Defining relation?
Answer in your set
Question (front)
Which common mistake happens with Cosine similarity between vectors?
Answer in your set
+ 7 more cards: units, variables, derivation, example, exam task
The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for cosSim = (x·y)/(‖x‖‖y‖): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential