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Vector Dot Product Calculator

Find the dot product of two three-dimensional vectors and the angle between them, with a step-by-step worked example showing every multiplication and sum.

Vector a

Vector b

Result

32

Vectors a = (1, 2, 3) and b = (4, 5, 6) have a dot product of 32

Quick Answer

The vector dot product multiplies corresponding components of two vectors and sums the results: a · b = a₁b₁ + a₂b₂ + a₃b₃. The result is always a single number, not a vector. For a = (1, 2, 3) and b = (4, 5, 6), the dot product is (1×4) + (2×5) + (3×6) = 4 + 10 + 18 = 32. The calculator also finds the angle between the two vectors in degrees as a secondary result.

Vector Dot Product Explained: What It Actually Measures

The vector dot product, also called the scalar product or Euclidean inner product, takes two vectors and returns a single number that captures how much the two vectors point in the same direction. Unlike the cross product, which returns a new vector perpendicular to the original two, the dot product returns a scalar. Unlike matrix multiplication, it works on two vectors of the same length and produces one number per pair, not a matrix. The dot product has two equivalent definitions. The algebraic definition sums the products of corresponding components, which is what this calculator uses directly. The geometric definition states that a · b = |a| × |b| × cos(θ), where θ is the angle between the two vectors. These two definitions are equal, and the calculator uses that equivalence to solve for the angle as a secondary output: once the dot product and both magnitudes are known, the angle follows from inverting the cosine.

The Vector Dot Product Formula And How It's Applied

a · b = a₁ × b₁ + a₂ × b₂ + a₃ × b₃. Angle between vectors: θ = arccos(a · b ÷ (|a| × |b|)), where |a| = √(a₁² + a₂² + a₃²).

  • Algebraic formula: a · b = a₁ × b₁ + a₂ × b₂ + a₃ × b₃.
  • Angle formula: θ = arccos(a · b ÷ (|a| × |b|)).
  • Magnitude: |a| = √(a₁² + a₂² + a₃²).
  • If a · b = 0, vectors are perpendicular (orthogonal).

Using This Vector Dot Product Calculator: Step By Step

Inputs

  • Components of vector a (a₁, a₂, a₃); negative values are accepted.
  • Components of vector b (b₁, b₂, b₃); negative values are accepted.

Steps

  1. Enter the three components of vector a in the a₁, a₂, a₃ fields.
  2. Enter the three components of vector b in the b₁, b₂, b₃ fields.
  3. Read the dot product from the primary result field.
  4. Read the angle between the vectors in degrees from the secondary result field.
  5. A dot product of zero confirms perpendicularity; no angle calculation is needed to see that relationship.

Vector Dot Product In Practice: A Full Worked Example

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

  1. Multiply components: 1×4 = 4, 2×5 = 10, 3×6 = 18.
  2. Sum products: a · b = 4 + 10 + 18 = 32.
  3. Find magnitudes: |a| = √14 = 3.742, |b| = √77 = 8.775.
  4. Find angle: cos(θ) = 32 ÷ (3.742 × 8.775) = 0.9747, θ = arccos(0.9747) = 12.9°.

Dot product = 32 (exact). Angle = 12.9 degrees.

When A Vector Dot Product Calculator Gives The Right Answer

Use the dot product calculator whenever you need to test whether two vectors are perpendicular, find the angle between two directions in 3D space, project one vector onto another (the dot product divided by the magnitude gives the projection length), or compute work done by a force in physics (force dot displacement equals work). In linear algebra, the dot product is the foundation for orthogonality checks, inner product spaces, and principal component analysis.

For other math calculations, visit the math calculators hub, and check the glossary for definitions of common technical terms.

Assumptions

  • Both vectors must be in the same coordinate system and have the same number of dimensions (3D for this calculator).
  • The angle output is in degrees in the range 0 to 180.
  • Component values are real numbers; complex vector components are not handled.
  • The calculator computes the standard Euclidean dot product, not a weighted inner product.

Limitations

  • Handles 3D vectors only; 2D vectors can be entered with a₃ = b₃ = 0.
  • Does not compute the cross product (which returns a vector perpendicular to both a and b); that is a separate operation.
  • Does not perform vector addition, subtraction, or scalar multiplication.
  • Not a matrix multiplication tool; for matrix operations, use a dedicated linear algebra calculator.

In Practice

The most common confusion around the dot product is mixing it up with the cross product. The dot product of two 3D vectors returns one number; if the number is zero, the vectors are perpendicular. The cross product of two 3D vectors returns a new third vector; if the vectors are parallel, the cross product is the zero vector. They answer different questions: the dot product tells you about the angle between two directions; the cross product tells you about the axis perpendicular to both. When a SERP shows "dot product" and "cross product" side by side, they are not interchangeable operations, and this calculator handles only the dot product.

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Vector Dot Product Calculator: Frequently Asked Questions

What Is the Dot Product of Two Vectors?

The dot product of two vectors is a single number calculated by multiplying corresponding components and summing the results: a · b = a₁b₁ + a₂b₂ + a₃b₃. Also called the scalar product or Euclidean inner product, it measures how much two vectors align with each other. A result of zero means the vectors are perpendicular; a positive result means they share a directional component; a negative result means they point in roughly opposite directions.

How Is the Dot Product Different From the Cross Product?

The dot product returns a scalar (one number) and measures the alignment between two vectors. The cross product returns a vector perpendicular to both input vectors and measures the area of the parallelogram they span. For two vectors a and b: a · b is a number; a × b is a new vector in 3D space. Use the dot product for angle and projection calculations; use the cross product for finding perpendicular directions and torque problems.

When Is the Dot Product Zero?

The dot product is zero when the angle between the two vectors is exactly 90 degrees, meaning the vectors are perpendicular (orthogonal). For example, the standard basis vectors i = (1, 0, 0) and j = (0, 1, 0) have a dot product of (1×0) + (0×1) + (0×0) = 0. The dot product test for perpendicularity is faster than computing the angle directly, making it a standard check in linear algebra and geometry algorithms.

Can I Use the Dot Product to Find the Angle Between Two 3D Lines?

Yes. Find a direction vector for each line, then apply the dot product formula: θ = arccos(a · b ÷ (|a| × |b|)). The angle returned is between 0 and 180 degrees. If you want the acute angle between two lines regardless of direction, take the absolute value of the dot product before dividing: arccos(|a · b| ÷ (|a| × |b|)).

Is the Dot Product the Same As the Scalar Product?

Yes. Dot product, scalar product, and Euclidean inner product are three names for the same operation. The term "scalar product" emphasises that the result is a scalar; "dot product" refers to the dot notation a · b; "Euclidean inner product" places it in the context of inner product spaces in linear algebra. All three mean the component-wise multiply-and-sum operation this calculator performs.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.