Cosine Similarity Calculator - Vector Score and Angle

A cosine similarity calculator measures how close two vectors point in the same direction. It returns a unitless score in [-1, 1] that equals 1 for parallel, 0 for orthogonal, and -1 for opposite vectors, plus the dot product, the magnitudes, and the angle in degrees and radians.

• • Information retrieval and document similarity: Score two text documents represented as bag-of-words or TF-IDF vectors to see whether they cover the same topic, without caring about document length.
• • Recommendation systems and embedding search: Compare a query embedding against a catalog of item embeddings to find the closest matches, since cosine similarity is the standard measure for dense vector search.
• • Linear algebra homework and sanity checks: Verify a hand-computed dot product, magnitudes, or angle for a textbook problem, including cases that come out to clean values like 0, 1, -1, or a 90-degree angle.

Cosine similarity ignores magnitude, so two vectors that point the same direction always score 1 even when one is much longer than the other, which is exactly why it is the standard for length-normalized text vectors.

The numerator of the formula is just the dot product, so when you only need that piece of the answer, the Dot Product Calculator page runs the same component multiplication and addition on a fresh pair of inputs.

The page implements the cosine similarity formula: the dot product of the two vectors divided by the product of their Euclidean magnitudes. It reads the first n components, computes the dot product, the two norms, and the angle, and clamps the final score to [-1, 1].

• A and B: The two input vectors, each with n real components.
• A . B: Dot product: sum of a_i * b_i. The numerator of the formula.
• ||A|| and ||B||: Euclidean magnitudes of A and B. Together they form the denominator.
• cos(theta): The cosine similarity, in [-1, 1]: 1 for parallel, 0 for orthogonal, -1 for opposite vectors.

The same formula works for any positive integer n, so the calculator accepts 2D through 5D inputs. The magnitudes in the denominator handle the normalization for you.

According to Wikipedia (Cosine similarity), the cosine similarity between two n-dimensional vectors A and B is defined as the dot product A·B divided by the product of their Euclidean norms, which equals the cosine of the angle between them.

The two magnitudes in the denominator come from the same arithmetic as a single-vector L2 norm, which the Vector Magnitude Calculator page computes directly when you only need the length of one vector.

Four short ideas explain why the formula looks the way it does and how to read the score you get back.

Once you know the dot product, the two magnitudes, and the angle, the cosine similarity follows from one division, and the rest of the page is just rounding and unit conversion.

If your data has a meaningful mean and you want to measure whether two series move together rather than whether two weight vectors point the same way, the Pearson Correlation Calculator page is the centred-data version of this same idea.

Five short steps cover everything from a clean 2D textbook example to a 5D embedding-like input.

1. 1 Pick a dimension: Choose 2, 3, 4, or 5. Only the first n components of each vector are used, so higher components can stay at 0 unless you actually need them.
2. 2 Enter the components of vector A: Type a1, a2, ... in order. Negative numbers, decimal numbers, and large numbers are all allowed.
3. 3 Enter the components of vector B: Type b1, b2, ... in the matching order. Components past n are ignored.
4. 4 Read the primary result: The cosine similarity appears in the large primary tile. The closer it is to 1, the more the two vectors point the same way; the closer to -1, the more they point opposite ways; 0 means orthogonal.
5. 5 Use the supporting values: Read the angle in degrees or radians to translate the score into a geometric angle, and inspect the dot product and magnitudes.

Try the 3D example A = (1, 2, 1) and B = (2, 1, 1). The calculator returns dot product = 5, ||A|| = ||B|| = sqrt(6) ~ 2.4495, similarity = 5/6 ~ 0.8333, and angle ~ 33.56 degrees (0.5857 rad). That is the kind of high-but-not-1 score that is typical for two related but not identical document embeddings.

When the two vectors are best read as point coordinates and you care about the actual distance between them instead of the angle, the 2D Distance Calculator page turns the same (x, y) pairs into a Euclidean distance with the step-by-step math.

These benefits matter most when you need a quick, transparent similarity score and do not want to re-derive the formula or write a script.

• • Transparent step-by-step math: The page shows the dot product, both magnitudes, the angle, and the similarity in one view, so you can see which arithmetic produced the score.
• • Works for 2D through 5D: Choose 2, 3, 4, or 5 components without reloading. The same formula runs on the first n components, matching how cosine similarity is used in text retrieval and embedding search.
• • Returns the angle in two units: Read the angle in degrees or radians. Degrees are 0 to 180, radians are 0 to pi.
• • Handles negative and decimal components: Negative weights, decimal weights, and large weights all use the same formula. The page does not require you to normalize the inputs first.
• • Catches the zero-vector edge case: If either vector is all zeros, the page shows a clear error explaining that cosine similarity is undefined when there is no direction to compare.

Use the page to confirm a homework answer, sanity-check an embedding search, or pre-validate a pair of vectors before you hand them to a longer script.

If you already have a cosine similarity score and only need the angle in degrees or radians, the cos⁻¹ (arccos) Calculator page runs arccos on a single number without re-typing the vectors.

The formula is the same in every case, but a few factors change how the score should be read and what it tells you about the inputs.

• • Cosine similarity ignores magnitude, so two vectors with very different lengths but the same direction will always score 1. When magnitude matters, use the 2D or 3D distance calculator on the underlying coordinates.
• • The formula returns a value in [-1, 1], not a probability. A score of 0.5 is not a 50% chance that two documents are duplicates; it just means the angle between the vectors is 60 degrees.
• • The result is meaningful only when both vectors are non-zero. The page blocks the calculation for zero vectors, but the underlying geometry still leaves the comparison undefined in that case.

According to scikit-learn, cosine_similarity computes the cosine of the angle between two vectors, returned as a similarity score in the range [-1, 1] that equals 1 for parallel vectors and -1 for opposite vectors, with the corresponding cosine distance defined as 1 minus this score.

When magnitude does matter for the same two 3D inputs and you want the actual length of the line between them rather than the angle, the 3D Distance Calculator page handles the (x, y, z) case directly.