How do you calculate the dot product of two vectors?

Multiply the two vectors component by component and add the results: A·B = a₁b₁ + a₂b₂ + … + aₙbₙ. Both vectors must have the same number of components. For example, [1,2,3]·[4,5,6] = 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32. The answer is a single number (a scalar), not another vector.

What does it mean when the dot product is zero?

A dot product of zero means the two vectors are orthogonal — perpendicular, at a 90° angle — provided neither is the zero vector. This is the standard test for perpendicularity in any number of dimensions. For example, [2,−1,3]·[1,5,1] = 2 − 5 + 3 = 0, so those two vectors meet at a right angle even though you cannot easily picture them.

How do you find the angle between two vectors using the dot product?

Rearrange the geometric identity A·B = ‖A‖‖B‖cosθ to get cosθ = (A·B)/(‖A‖·‖B‖), then take the inverse cosine: θ = arccos(cosθ). Each magnitude ‖A‖ is the square root of the sum of squares of its components. The calculator clamps the cosine to [−1, 1] first so floating-point rounding never produces an invalid arccos.

Is the dot product a scalar or a vector?

The dot product is a scalar — a single number — which is why it is also called the scalar product. That is what separates it from the cross product, which returns a vector. The dot product tells you how much two vectors point in the same direction; its sign alone says whether the angle between them is acute (positive), right (zero), or obtuse (negative).

What is the difference between the dot product and the cross product?

The dot product takes two vectors and returns a scalar (A·B = ‖A‖‖B‖cosθ); it works in any dimension and measures alignment. The cross product takes two 3-dimensional vectors and returns a third vector perpendicular to both, with length ‖A‖‖B‖sinθ; it measures the area they span and only exists in 3-D (and 7-D). This tool computes the dot product; a separate page would be needed for the cross product.

Can the dot product be negative?

Yes. The dot product is negative whenever the angle between the vectors is obtuse — greater than 90° — because the cosine of an obtuse angle is negative. For example, [1,2,3]·[−2,−4,−6] = −28, and those vectors point in exactly opposite directions (180°). A positive dot product means an acute angle, and zero means a right angle.

Do the two vectors need to be the same length?

Yes. The dot product pairs components by position — a₁ with b₁, a₂ with b₂, and so on — so both vectors must have the same number of components. A 2-dimensional vector has no defined dot product with a 3-dimensional one. This calculator rejects mismatched lengths with a clear message naming both counts rather than padding or truncating, which would give a misleading answer.

Does this calculator send my data anywhere?

No. Parsing your numbers, multiplying the pairs, summing them, and taking the square roots and arccos all happen in your browser with plain JavaScript. Nothing is uploaded, logged, or stored, so you can paste real embedding values or coursework figures without concern. The page also keeps working offline once it has loaded.

AI · Linear algebra

Dot Product Calculator

Find the dot product of two vectors in your browser. See the scalar result, each vector's magnitude, the cosine, and the angle between them — with the full element-wise working behind every number. No signup, nothing uploaded.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jun 10, 2026

Dot product calculator

Vector A

Numbers separated by commas, spaces, or new lines.

Vector B

Must have the same number of components as A.

Examples

Dot product A · B

32.0000

Magnitudes

‖A‖ 3.7417

‖B‖ 8.7750 · 3-D

Angle

12.93°

0.2257 rad

Relationship

Acute angle (θ < 90°)

cosθ = 0.9746

Decimals

Angle

Cross-check. The direct sum Σ aᵢbᵢ gives 32.0000; the independent polarisation identity (‖A+B‖² − ‖A‖² − ‖B‖²)/2 — which never multiplies aᵢ by bᵢ — gives 32.0000. They reconcile, as they must.

Step-by-step working

Component	aᵢ	bᵢ	aᵢ·bᵢ	aᵢ²	bᵢ²
#1	1.0000	4.0000	4.0000	1.0000	16.0000
#2	2.0000	5.0000	10.0000	4.0000	25.0000
#3	3.0000	6.0000	18.0000	9.0000	36.0000
Totals			32.0000	14.0000	77.0000

A · B = Σ aᵢbᵢ = 32.0000

‖A‖ = √14.0000 = 3.7417

‖B‖ = √77.0000 = 8.7750

cosθ = 32.0000 / (3.7417 × 8.7750) = 0.9746

Method: A·B = Σ aᵢbᵢ; ‖A‖ = √(Σ aᵢ²); cosθ = (A·B)/(‖A‖‖B‖); θ = arccos(cosθ) — Wolfram MathWorld Dot Product and L2-Norm. Nothing leaves this page.

How it works

The dot product (also called the scalar product or inner product) takes two vectors of the same length and returns a single number. It is the sum of the products of matching components, and it underpins cosine similarity, projections, and most of the geometry used in machine learning. The definitions below follow Wolfram MathWorld and standard linear algebra.

For two equal-length vectors A = [a₁…aₙ] and B = [b₁…bₙ]:

A · B = Σ aᵢbᵢ = a₁b₁ + a₂b₂ + … + aₙbₙ

The tool computes the result and its geometry in four steps:

Dot product. Multiply the vectors component by component and add the results: A · B = Σ aᵢbᵢ. This single number is the scalar product.
Magnitudes.Take the square root of each vector's sum of squares: ‖A‖ = √(Σ aᵢ²) and likewise for ‖B‖ — the Euclidean magnitude, or L2 norm.
Cosine of the angle. Rearranging the geometric identity A · B = ‖A‖‖B‖cosθ gives cosθ = (A·B)/(‖A‖·‖B‖). This is defined only when both magnitudes are non-zero; a zero vector has no direction, so the tool shows a clear note instead of dividing by zero.
Angle. The angle is θ = arccos(cosθ), shown in degrees and radians. The cosine is clamped to [−1, 1] first so floating-point drift never produces an invalid arccos.

The sign of the dot product alone tells you the kind of angle: positive means acute (the vectors broadly agree), zero means orthogonal (perpendicular, a right angle), and negative means obtuse, up to exactly opposite at 180°. As a credibility check, the calculator also recovers the dot product a second, independent way — the polarisation identity A·B = (‖A+B‖² − ‖A‖² − ‖B‖²)/2, which never multiplies aᵢ by bᵢ — and confirms the two routes agree. The dot product is the numerator of cosine similarity, so the cosine similarity and Euclidean distance tools linked below sit in the same family.

Worked examples

General 3-D vectors — A = [1, 2, 3], B = [4, 5, 6]

Dot product: 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32
‖A‖ = √(1 + 4 + 9) = √14 = 3.741657
‖B‖ = √(16 + 25 + 36) = √77 = 8.774964
cosθ = 32 / (3.741657 × 8.774964) = 32 / 32.832910 = 0.974632
θ = arccos(0.974632) = 0.225726 rad = 12.93° → Acute angle

Orthogonal vectors — A = [2, −1, 3], B = [1, 5, 1]

Dot product: 2·1 + (−1)·5 + 3·1 = 2 − 5 + 3 = 0
‖A‖ = √(4 + 1 + 9) = √14, ‖B‖ = √(1 + 25 + 1) = √27
cosθ = 0 / (√14 × √27) = 0
θ = arccos(0) = 90.00°
Dot product = 0 → Orthogonal (perpendicular)

Opposite direction — A = [1, 2, 3], B = [−2, −4, −6] (B = −2·A)

Dot product: 1·(−2) + 2·(−4) + 3·(−6) = −2 − 8 − 18 = −28
‖A‖ = √14, ‖B‖ = √56, product = √784 = 28
cosθ = −28 / 28 = −1
θ = arccos(−1) = 180.00°
Negative dot product, cosθ = −1 → Opposite direction (anti-parallel)

Frequently asked questions

Sources & references

The formulas on this page were last cross-checked against these sources on 2026-06-10. The dot product is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.

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