Updated Vector & Similarity Tool

Cosine Similarity Calculator

Compute cosine similarity and angle between two vectors in any dimension. Enter components or coordinates to get cosine similarity, angle in degrees, dot product, magnitudes and a distance score based on 1 − cosine.

Vector Geometry Angle Between Vectors Similarity Measure Any Dimension

Calculate Cosine Similarity Between Two Vectors

This Cosine Similarity Calculator lets you compare two vectors either by entering their components directly or by specifying 2D/3D coordinates. It returns the cosine similarity (from −1 to 1), the angle in degrees, dot product, magnitudes of each vector and a simple distance score based on 1 − cosine similarity.

Cosine similarity between vectors u and v is defined as cos(θ) = (u · v) / (‖u‖ ‖v‖). It measures how aligned the two vectors are, independent of their overall length. A value of 1 means they point in the same direction, 0 means they are orthogonal and −1 means they point in exactly opposite directions.

Enter components separated by commas, spaces or new lines.
Must have the same dimension as vector u.

This mode works for any dimension (2D, 3D, 100D and beyond). It is suitable for both geometric vectors and high-dimensional feature or embedding vectors.

Leave z empty for 2D vectors.
Both points must use the same dimension (either 2D or 3D).

In this mode, each point is treated as a vector from the origin. The calculator computes cosine similarity and angle between these position vectors.

Cosine Similarity Calculator – Measure Alignment Between Vectors

Cosine similarity is a popular measure for comparing the direction of two vectors. Unlike Euclidean distance, which depends on absolute magnitude, cosine similarity focuses on how aligned the vectors are. The Cosine Similarity Calculator from MyTimeCalculator makes it easy to compute the cosine similarity, angle, dot product and magnitudes for vectors in any dimension.

This is useful both in classical vector geometry (2D and 3D vectors) and in modern applications such as text embeddings, recommendation systems, feature vectors in machine learning and high-dimensional data analysis.

1. Formula for Cosine Similarity

For two vectors u and v in n-dimensional space, the cosine similarity is defined as:

cos(θ) = (u · v) / (‖u‖ ‖v‖)

where:

  • u · v is the dot product of u and v.
  • ‖u‖ is the magnitude (Euclidean norm) of vector u.
  • ‖v‖ is the magnitude of vector v.
  • θ is the angle between the two vectors.

The result is always between −1 and 1:

  • cos(θ) = 1 → vectors point in exactly the same direction.
  • cos(θ) = 0 → vectors are orthogonal (perpendicular).
  • cos(θ) = −1 → vectors point in exactly opposite directions.

2. Dot Product and Magnitude

The calculator also shows the intermediate quantities that appear in the cosine similarity formula:

  • Dot product: For vectors u = (u1, …, un) and v = (v1, …, vn),
    u · v = Σ ui vi.
  • Magnitude (norm): ‖u‖ = √(u12 + … + un2), and similarly for ‖v‖.

If either vector has zero magnitude, cosine similarity is undefined because you cannot divide by zero. The calculator detects this case and reports that the similarity is not defined.

3. Cosine Distance Based on 1 − Similarity

In addition to cosine similarity, the calculator reports a simple cosine distance:

dcos(u, v) = 1 − cos(θ)

When similarity is 1 (perfectly aligned), the distance is 0. When similarity is 0 (orthogonal), the cosine distance is 1. When similarity is negative, the cosine distance is greater than 1. While cosine distance is not a true metric in all settings, it is widely used as a convenient dissimilarity measure derived from cosine similarity.

4. How to Use the Cosine Similarity Calculator

  1. Choose a tab. Use the Vector Components tab if you want to enter component lists like (1, 2, 3). Use the 2D / 3D Points tab if you prefer entering coordinates for points A and B in 2D or 3D.
  2. Enter your data. For vectors, paste the components using commas, spaces or line breaks. For points, fill in x and y (and optionally z) for each point. Both inputs must have the same dimension.
  3. Click the calculate button. The calculator will compute cosine similarity, angle in degrees, dot product, magnitudes and cosine distance, and display a short verbal interpretation.
  4. Interpret the result. Values close to 1 indicate strong alignment, values near 0 indicate little or no directional similarity, and negative values indicate opposite directions.
  5. Try multiple vectors. Adjust the inputs to compare different vectors or to see how scaling a vector (multiplying by a constant) does not change its cosine similarity to another vector.

5. Applications of Cosine Similarity

  • Text and embeddings: Measuring similarity between document or sentence embeddings in information retrieval and search systems.
  • Recommendation systems: Comparing user or item feature vectors to recommend similar items.
  • Clustering and classification: Using cosine similarity as a similarity measure in clustering algorithms for high-dimensional data.
  • Vector geometry: Finding the angle between forces, velocities or other physical vectors in physics and engineering problems.
  • Data analysis: Comparing normalized feature vectors when absolute magnitude is not important but direction andative composition are.

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Cosine Similarity Calculator FAQs

Frequently Asked Questions

Quick answers to common questions cosine similarity, angles between vectors, dot products and how to interpret the calculator outputs.

A cosine similarity of 1 means the vectors point in exactly the same direction (θ = 0°). A value of 0 means they are orthogonal (θ = 90°), sharing no directional component. A value of −1 means they point in exactly opposite directions (θ = 180°). Values in between capture varying degrees of alignment or opposition between the two vectors.

In the cosine similarity formula, both vectors are divided by their magnitudes. If you scale a vector by a positive constant, its direction stays the same and the scale factor cancels out in the ratio. This means cosine similarity depends only on direction, not on overall length, making it ideal for comparing patterns or composition rather than absolute size.

If a vector has all components equal to zero, its magnitude is also zero and the cosine similarity formula involves division by zero. In this case, cosine similarity is undefined. The calculator detects this and reports that similarity cannot be computed when one or both vectors are zero vectors, since they have no meaningful direction.

Euclidean distance measures the straight-line distance between two points, which depends on both direction and magnitude. Cosine similarity focuses only on the angle between vectors, ignoring overall length. Two vectors that are multiples of each other have zero Euclidean distance only if they are identical, but they have cosine similarity 1 as long as they point in the same direction, regardless of scale. Which measure is better depends on whether magnitude matters for your problem.

Yes. The Vector Components tab supports any dimension, as long as both vectors have the same number of components. You can paste long component lists, and the calculator will compute the dot product, norms and cosine similarity numerically. This makes it suitable for comparing feature vectors, document embeddings or other high-dimensional representations used in data science and machine learning workflows.

Due to finite precision in floating-point arithmetic, tiny rounding errors can occur when computing dot products and magnitudes. This may occasionally produce results like 1.0000001 instead of exactly 1. The calculator clamps the cosine value into the valid range [−1, 1] before computing the angle, but you may still see very small deviations in printed values. These are normal numerical effects and not conceptual errors in the formula.

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