What is the dot product formula?

For two n-dimensional vectors A = (a1, a2, …, an) and B = (b1, b2, …, bn), the dot product is A·B = a1·b1 + a2·b2 + … + an·bn. It is also equal to |A||B|cos θ, where θ is the angle between the vectors and |A|, |B| are their magnitudes.

How do I find the angle between two vectors?

Rearrange the geometric form of the dot product: cos θ = (A·B) / (|A| × |B|), then take the inverse cosine (arccos). The result is always between 0° and 180°. This calculator does all three steps automatically and shows the full substitution.

What does a negative dot product mean?

A negative dot product means the angle between the vectors is greater than 90° (obtuse). The two vectors point in "generally opposite" directions. A dot product of exactly zero means the vectors are perpendicular (orthogonal). A positive dot product means the angle is less than 90° (acute).

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

The dot product of two vectors produces a scalar (a single number) and measures how much one vector aligns with another. The cross product is only defined in 3D and produces a new vector that is perpendicular to both inputs; its magnitude equals the area of the parallelogram spanned by the two vectors. This calculator computes both when you are in 3D mode.

What is a scalar projection?

The scalar projection of A onto B is compA(B) = A·B / |B|. It tells you how long the "shadow" of A is when cast along the direction of B. Scalar projection is used in physics to find the component of a force along a given direction, and in machine learning for feature similarity.

When is the dot product used in the real world?

The dot product is fundamental in physics (work = F·d), computer graphics (lighting via the Lambertian shading model), machine learning (cosine similarity for text embeddings), robotics (joint torque computation), and signal processing (correlation and matched filters). Any time you need to measure the alignment or overlap between two directional quantities, the dot product is the tool.

Dot Product Calculator

The dot product (also called the scalar product or inner product) is one of the most widely used operations in mathematics, physics, and computer science. Given two vectors A and B, it produces a single scalar that encodes both the magnitudes of the vectors and the cosine of the angle between them.

The formula

For vectors in n dimensions:

A · B = a₁b₁ + a₂b₂ + … + aₙbₙ

The equivalent geometric form links the dot product to the angle θ between the vectors:

A · B = |A| |B| cos θ

Rearranging gives the angle formula, which is how this calculator finds θ:

θ = arccos( A · B / (|A| |B|) )

where |A| = √(a₁² + a₂² + … + aₙ²) is the Euclidean magnitude of A.

How it works

Dot product — component-wise multiplication followed by summation. The calculator shows every multiplicative term before summing so you can verify each step.
Magnitudes — computed as the square root of the sum of squared components (the L² norm).
Angle — derived from arccos(dot / (|A| × |B|)). The result is clamped to [-1, 1] before taking arccos to guard against floating-point rounding errors.
Scalar projections — the component of one vector along the other: compA→B = A·B / |B| and compB→A = A·B / |A|.
Cross product (3D only) — A × B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx). Its magnitude equals the area of the parallelogram spanned by A and B.

The geometric flags (orthogonal, parallel, anti-parallel, acute, obtuse) are derived automatically: two vectors are orthogonal when |dot| is below 10⁻¹⁰ (within floating-point tolerance), and parallel or anti-parallel when |cos θ| ≥ 1 − 10⁻⁹.

Worked example

Consider A = (3, 1, −2) and B = (4, −3, 1) in 3D.

Step 1 — dot product:

A · B = (3)(4) + (1)(−3) + (−2)(1) = 12 − 3 − 2 = 7

Step 2 — magnitudes:

|A| = √(9 + 1 + 4) = √14 ≈ 3.7417

|B| = √(16 + 9 + 1) = √26 ≈ 5.0990

Step 3 — angle:

cos θ = 7 / (3.7417 × 5.0990) ≈ 7 / 19.079 ≈ 0.3670

θ = arccos(0.3670) ≈ 68.49°

Step 4 — cross product:

A × B = ((1)(1)−(−2)(−3), (−2)(4)−(3)(1), (3)(−3)−(1)(4)) = (1−6, −8−3, −9−4) = (−5, −11, −13)

|A × B| = √(25 + 121 + 169) = √315 ≈ 17.748 (area of the parallelogram)

Enter these values into the calculator above to verify each step in the working panel.

Vectors	Dot product	Angle
A=(1,0), B=(0,1)	0	90° (orthogonal)
A=(1,0), B=(1,0)	1	0° (parallel)
A=(3,4), B=(−4,3)	0	90° (orthogonal)
A=(3,1,−2), B=(4,−3,1)	7	≈ 68.49°
A=(1,2,3), B=(−3,−2,−1)	−10	≈ 135.58° (obtuse)