Dot Product Calculator

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Walk-through

How to Use This Calculator

3 steps

1

Choose 2D or 3D and enter both vectors

Select 2D (x, y) or 3D (x, y, z), then type the components of vector a and vector b into the labelled fields. You can use positive or negative numbers and decimals. The calculator updates every result instantly as you type.

2

Read the dot product, angle, and orthogonality

The Dot Product tab shows a · b = Σaᵢbᵢ. The Angle Between tab shows the angle θ = arccos((a · b) / (|a||b|)) in degrees. The Orthogonal Check tab tells you whether a · b = 0, meaning the vectors meet at a right angle.

3

Try a few vector pairs

Use the example chips to load a 2D pair, a 3D pair, or a known orthogonal pair (like (1,0) and (0,1)). Comparing a parallel pair, a perpendicular pair, and an arbitrary pair builds intuition for how the dot product's sign and magnitude relate to the angle between vectors.

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Reference

Formula & Methodology

3 formulas

Dot product

a · b = Σ aᵢbᵢ = a₁b₁ + a₂b₂ + ... + aₙbₙ

The dot product (also called the scalar product) of two vectors is the sum of the products of their corresponding components. For 2D vectors it's axbx + ayby; for 3D vectors it's axbx + ayby + azbz. The result is always a single scalar number, not a vector.

Angle between two vectors

θ = arccos( (a · b) / (|a| |b|) )

Rearranging the geometric definition of the dot product (a · b = |a||b|cos θ) gives the angle between the two vectors. |a| and |b| are the vectors' magnitudes (Euclidean lengths). This formula is undefined when either vector has zero magnitude — there is no direction to measure an angle from.

Orthogonality test

a ⊥ b ⇔ a · b = 0

Two non-zero vectors are orthogonal (perpendicular, meeting at 90°) exactly when their dot product equals zero. This follows directly from the angle formula, since cos(90°) = 0.

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Glossary

Key Terms Explained

6 terms

Dot product A scalar value computed by summing the products of the corresponding components of two vectors of the same dimension. Also called the scalar product.

Scalar product Another name for the dot product — emphasizes that the result is a single number (scalar), unlike the cross product, which returns a vector.

Angle between vectors The angle θ, between 0° and 180°, formed where two vectors' tails meet when drawn from the same origin. Computed from the dot product and both vectors' magnitudes.

Orthogonal Two vectors are orthogonal when they meet at exactly 90°, which happens if and only if their dot product is 0. In 2D and 3D this is the same as 'perpendicular.'

Magnitude The length of a vector, written |v|, computed as the square root of the sum of the squares of its components (the Euclidean norm).

Vector components The individual coordinate values that make up a vector — x and y for 2D vectors, or x, y, and z for 3D vectors.

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Scenarios

Real-World Examples

3 worked examples

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2D vectors

a = (1, 2), b = (3, 4)

Vector a (1, 2) Vector b (3, 4)

a · b = (1)(3) + (2)(4) = 3 + 8 = 11. Since the dot product is positive, the angle between a and b is less than 90° (here it's about 10.3°) — the vectors point in roughly the same general direction.

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3D vectors

a = (1, 2, 3), b = (4, 5, 6)

Vector a (1, 2, 3) Vector b (4, 5, 6)

a · b = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32. |a| ≈ 3.742 and |b| ≈ 8.775, so θ = arccos(32 / (3.742 × 8.775)) ≈ 12.9° — the vectors point in nearly the same direction.

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Orthogonal case

a = (1, 0), b = (0, 1)

Vector a (1, 0) Vector b (0, 1)

a · b = (1)(0) + (0)(1) = 0. Because the dot product is 0, a and b are orthogonal — they form a 90° angle, exactly as you'd expect from the x-axis and y-axis unit vectors.

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Deep Dive

Understanding the Dot Product Calculator

3 chapters · 4 min read

The dot product is one of the most useful operations in vector algebra: it turns two vectors into a single number that tells you how much they point in the same direction. This calculator computes a · b for 2D or 3D vectors, converts that result into the angle between the vectors, and flags when the vectors are orthogonal.

How the Dot Product Calculator works

Given two vectors a and b of the same dimension, the calculator sums the products of their matching components: a · b = a_xb_x + a_yb_y (2D) or a_xb_x + a_yb_y + a_zb_z (3D). It then computes each vector's magnitude (|a| and |b|) and uses the identity a · b = |a||b|cos θ to solve for the angle θ = arccos((a · b) / (|a||b|)). Finally, it checks whether a · b equals zero (within floating-point tolerance) to report orthogonality.

Inputs and what they mean

Pick 2D or 3D from the dimension selector, then enter the x, y (and z, for 3D) components of vector a and vector b. Components can be positive, negative, zero, or decimals. There is no unit conversion — the result is in the same units as your inputs (e.g., if your components are in meters, the dot product and magnitudes are in meter-based units, and the angle is always reported in degrees). The sign of the dot product is informative on its own: positive means the vectors generally point the same way (angle under 90°), negative means they generally point opposite ways (angle over 90°), and zero means they're exactly perpendicular.

Limits and edge cases

The angle between vectors is undefined when either vector has zero magnitude — there is no direction to measure from a zero vector, so the Angle Between and Orthogonal Check tabs will show a message instead of a number in that case. This calculator handles 2D and 3D vectors; for higher-dimensional (nD) dot products, the same Σaᵢbᵢ formula applies, just extended with more terms — you can compute those by hand or with a general-purpose math tool. Floating-point rounding means the orthogonality check uses a small numerical tolerance rather than requiring the dot product to be exactly 0.

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Questions

Frequently Asked Questions

6 questions

What is the formula for the dot product?+

The dot product of two vectors a and b is a · b = Σaᵢbᵢ — the sum of the products of their corresponding components. For 2D vectors that's axbx + ayby; for 3D vectors it's axbx + ayby + azbz. The result is always a single number (a scalar), not a vector.

What is the formula for the angle between two vectors?+

θ = arccos((a · b) / (|a| |b|)), where |a| and |b| are the magnitudes (lengths) of the two vectors. This comes from rearranging the geometric identity a · b = |a||b|cos θ.

How do I know if two vectors are orthogonal?+

Two vectors are orthogonal (perpendicular) exactly when their dot product equals 0. This calculator's Orthogonal Check tab computes a · b and reports Yes or No automatically.

Does this work for 2D and 3D vectors?+

Yes — switch the Dimensions selector to 2D (x, y) or 3D (x, y, z). The same Σaᵢbᵢ formula extends to any number of dimensions (nD); for higher dimensions, add one more axᵢbᵢ term per dimension.

Is the dot product commutative?+

Yes. a · b = b · a, because multiplication of the individual scalar components is commutative and the sum doesn't depend on the order of the terms.

What does a negative dot product mean?+

A negative dot product means the angle between the two vectors is greater than 90° — they generally point in opposing directions. A positive dot product means the angle is less than 90° (pointing roughly the same way), and a dot product of exactly 0 means the vectors are orthogonal (exactly 90°).

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Enter vectors a and b

a · b

Angle Between Vectors

Orthogonal Check

How to Use This Calculator

Choose 2D or 3D and enter both vectors

Read the dot product, angle, and orthogonality

Try a few vector pairs

Formula & Methodology

Key Terms Explained

Real-World Examples

2D vectors

3D vectors

Orthogonal case

Understanding the Dot Product Calculator

How the Dot Product Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Dot Product Calculator

Enter vectors a and b

a · b

Angle Between Vectors

Orthogonal Check

How to Use This Calculator

Choose 2D or 3D and enter both vectors

Read the dot product, angle, and orthogonality

Try a few vector pairs

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Dot Product Calculator

How the Dot Product Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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Guides & articles