Cosine Similarity Calculator Online

Calculator Input

Enter equal length vectors. You may separate values with commas, spaces, semicolons, pipes, or line breaks.

Vector A

Example: 3, 4, 5, 6

Vector B

Example: 2, 3, 5, 8

Optional Weights

Used only in weighted mode.

Calculation Mode

Decimal Precision

Quick Example

Formula Used

Cosine similarity:

cos(θ) = (A · B) / (||A|| × ||B||)

Dot product:

A · B = Σ(aᵢ × bᵢ)

Vector norm:

||A|| = √Σ(aᵢ²)

Weighted mode:

cos(θ) = Σ(wᵢaᵢbᵢ) / √Σ(wᵢaᵢ²) × √Σ(wᵢbᵢ²)

Cosine distance:

distance = 1 - cosine similarity

How to Use This Calculator

Enter the first vector in the Vector A box.
Enter the second vector in the Vector B box.
Keep both vectors the same length.
Select standard, centered, or weighted mode.
Add weights only when weighted mode is selected.
Choose the decimal precision for cleaner output.
Press the calculate button.
Review the result, chart, and component table.
Use the CSV or PDF button to save your work.

Example Data Table

Case	Vector A	Vector B	Mode	Expected Meaning
Similar direction	3, 4, 5, 6	2, 3, 5, 8	Standard	High positive similarity
Opposite direction	1, 2, 3	-1, -2, -3	Standard	Similarity near -1
Weighted profile	5, 2, 8, 1	4, 3, 7, 2	Weighted	Important dimensions affect more

Cosine Similarity Guide

What Cosine Similarity Means

Cosine similarity measures how close two vectors point in the same direction. It ignores absolute size and focuses on orientation. A score near 1 means both vectors have a similar pattern. A score near 0 means the vectors are mostly unrelated. A score near -1 means they point in opposite directions.

Why This Calculator Helps

Manual vector comparison can be slow. Each component must be multiplied, summed, squared, and checked. This calculator keeps those steps visible. It reports dot product, norms, cosine score, angle, cosine distance, Euclidean distance, Manhattan distance, and a percent style similarity. It also supports optional weights. Weights are useful when some dimensions are more important than others.

Common Uses

Cosine similarity is widely used in math, search, data science, and text analysis. It can compare documents after words are converted into numeric vectors. It can compare product features, user preferences, recommendation profiles, and signal shapes. The method is popular because two large vectors can still match well when their direction is alike.

Interpreting Results

A high positive score shows strong alignment. A low positive score suggests weak alignment. A negative score means the vectors move in opposite ways. The angle gives a geometric view. Smaller angles mean stronger similarity. Cosine distance is simply one minus the similarity score. It is helpful when a distance style result is needed.

Better Input Practice

Use equal vector lengths. Keep units consistent across matching positions. Avoid all-zero vectors because they have no direction. Use centered mode when the shared trend matters more than raw level. Use weighted mode when a few components should influence the result more strongly. Review the component table to see which dimensions drive the final score.

A Practical Check

Before trusting a score, inspect the values. Very large components can dominate the dot product. Missing values should be replaced with a consistent rule. Negative values are allowed, but they change interpretation. The chart helps reveal unusual dimensions. The export buttons let you save results for reports, audits, classroom work, or later comparison with other vector pairs. This makes the output easier to explain, repeat, and compare during later reviews with less confusion.

FAQs

1. What is cosine similarity?

Cosine similarity measures the angle-based similarity between two vectors. It compares direction, not size. A value near 1 means strong similarity. A value near 0 means weak relation. A value near -1 means opposite direction.

2. Can cosine similarity be negative?

Yes. Negative values appear when vectors point in opposite directions. This often happens when components have mixed signs or inverse patterns. A score of -1 means perfect opposite alignment.

3. Why must vectors have equal length?

Each component in one vector must match a component in the other vector. The calculator multiplies matching positions. Unequal lengths break the dot product and produce invalid comparison.

4. What does centered mode do?

Centered mode subtracts each vector mean before calculation. This reduces level effects and focuses on pattern movement. It is useful when relative shape matters more than raw magnitude.

5. When should I use weighted mode?

Use weighted mode when some dimensions are more important. Larger weights increase the influence of matching components. Use zero weight to ignore a component without deleting it.

6. What is cosine distance?

Cosine distance equals one minus cosine similarity. Smaller distance means stronger similarity. It is useful in clustering, search ranking, recommendation systems, and nearest neighbor tasks.

7. Can I compare text documents with this calculator?

Yes, if the documents are already converted into numeric vectors. These vectors may come from word counts, TF-IDF scores, embeddings, or other text representation methods.

8. Why is an all-zero vector invalid?

An all-zero vector has no direction. Its norm is zero, which makes division impossible in the cosine formula. Add meaningful values before calculating similarity.