Enter two vectors and compare directions in seconds. See dot product, angle hints, and similarity. Download clean reports and keep your calculations organized always.
| Vector A | Vector B | Dot product | Interpretation |
|---|---|---|---|
| [1, 2, 3] | [4, 5, 6] | 32 | Positive value, generally pointing in similar directions. |
| [3, -2] | [7, 1] | 19 | Positive value, not perpendicular, some alignment exists. |
| [2, 0, -1, 4] | [1, 5, 2, -3] | -12 | Negative value, tends toward opposite directions overall. |
Use these examples to verify your inputs and expected outputs.
For two vectors A = (a₁, a₂, …, aₙ) and B = (b₁, b₂, …, bₙ), the dot product is:
A · B = a₁b₁ + a₂b₂ + … + aₙbₙ
Magnitude is computed as |A| = √(a₁² + a₂² + … + aₙ²). When both magnitudes are non‑zero, the angle θ between vectors is derived from:
cos(θ) = (A · B) / (|A| |B|)
The dot product compresses two vectors into a single scalar that quantifies alignment. For A=(a1…an) and B=(b1…bn), the calculator multiplies matching components and sums them: a1b1+a2b2+…+anbn. This works for 2D, 3D, or any n you provide, so it fits coordinate geometry, numerical methods, and feature‑vector scoring in the same workflow.
A positive dot product indicates the vectors generally point in a similar direction, while a negative value suggests opposition. When the value is near zero and neither vector is zero‑length, the vectors are close to perpendicular. Because the operation is linear, doubling one vector doubles the dot product. In applied problems, treat units consistently, since mixing meters and centimeters can change results by orders of magnitude.
With magnitudes |A| and |B|, the tool also estimates the angle using cos(θ)=(A·B)/(|A||B|) and reports θ in degrees. Cosine similarity is the same ratio and ranges from −1 to 1, which makes it ideal for comparing text embeddings, user profiles, or signals. Minor rounding can push the ratio slightly beyond the valid range, so the calculator clamps it before acos for stable angles.
Accurate computation starts with clean inputs. Enter components separated by commas or spaces, and ensure both vectors have the same number of components before submitting. Scientific notation like 1.2e-3 is accepted for very small values. If a vector is all zeros, its magnitude becomes zero and the angle is undefined; the result area shows dashes instead of generating misleading degrees or similarity values.
Dot products appear across disciplines because they connect geometry and measurement. In geometry, A·B=0 is a fast orthogonality test and supports projections and component breakdowns. In physics, work equals force·displacement and depends on the angle, so the same force does less work when applied sideways. In data science, dot products drive linear models and retrieval ranking, enabling fast similarity checks over large datasets. For quality control, compare a hand‑computed example to confirm separators, signs, and rounding behave in real projects more reliably.
It measures how strongly two vectors align. It is used for projections, orthogonality tests, computing work in physics, and similarity scoring in analytics.
Yes. Component‑wise multiplication requires both vectors to have the same number of components. If dimensions differ, the calculator will show an error.
If both vectors have non‑zero magnitude, a dot product of zero implies perpendicular directions. If either vector is the zero vector, the dot product is zero but direction is undefined.
Yes. Inputs like 3.5e2 or 1.2e-3 are accepted, which helps when working with very large or very small components.
The calculator uses cos(θ)=(A·B)/(|A||B|) and then converts θ to degrees. If either magnitude is zero, the angle cannot be computed.
They include your original vector inputs, dimension count, dot product, magnitudes, cosine similarity, angle in degrees, and a timestamp of when the report was generated.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.