Dot product calculator

A dot product calculator multiplies two vectors to give a single number: multiply the matching components and add them. For a = (a₁, a₂) and b = (b₁, b₂), a · b = a₁b₁ + a₂b₂. Our Vector Calculator gives the dot product in 2D and 3D, and the angle between the vectors, which is what the dot product is mostly used for.

Choose Two Vectors mode for 2D, or 3D mode for vectors with a z component. It opens in Single mode, which has only one vector, so switching mode is the first step. The dot product is also called the scalar product.

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How to use the dot product calculator

1. Under Mode, choose Two Vectors for 2D or 3D.
2. In Vectors, type the components of a and b, or drag the arrows on the grid.
3. Read the dot product, a · b, and the angle θ between the vectors in Results.
4. Open Formulas to see the dot product and the angle formula with your numbers in.

Angles are in degrees unless you switch Degrees off under Display. 3D mode also gives each vector's magnitude and the cross product.

The dot product formula

In components, add the products of matching components:

Geometrically, the dot product is the product of the lengths and the cosine of the angle between them, which rearranges to give the angle:

Worked examples you can check in the calculator

Example 1: dot product and angle in 2D

Find a · b and the angle between a = (4, 1) and b = (2, 5).

The magnitudes are |a| = √17 and |b| = √29, so cos θ = 13 ÷ √(17 × 29) = 0.5855 and θ = 54.16°. In Two Vectors mode, type both vectors: Results shows a · b = 13 and θ = 54.16°.

Example 2: perpendicular vectors

For a = (3, −2) and b = (4, 6), a · b = 12 − 12 = 0. A dot product of 0 means the vectors are perpendicular, and the calculator shows θ = 90°. This is the quickest test for right angles between vectors.

Example 3: an obtuse angle

For a = (1, 2) and b = (−3, −1), a · b = −3 − 2 = −5. A negative dot product means the angle is obtuse, more than 90°: here it is 135°.

Example 4: 3D vectors

Find the angle between a = (2, −1, 3) and b = (4, 2, 1).

|a| = √14 and |b| = √21, so cos θ = 9 ÷ √294 = 0.5249 and θ = 58.34°. In 3D mode the calculator shows a · b = 9 and θ = 58.34°.

Check you have the formula:

What the sign of the dot product tells you

Where the dot product is used

• Work done in physics: W = F · d. A force of (3, 4) N moving an object (5, 0) m does 3 × 5 + 4 × 0 = 15 J of work.
• Projection: the length of a in the direction of b is a · b ÷ |b|. For Example 1 that is 13 ÷ √29 = 2.41.
• Checking perpendicular lines and planes in geometry and mechanics.

For a vector at right angles to both a and b, use the cross product instead: see our cross product calculator guide. The vector calculator guide covers the rest of the tool.

Common mistakes

• Writing the answer as a vector. The dot product is a scalar, a single number.
• Multiplying the wrong components. Multiply x with x, y with y and z with z, then add.
• Dropping negative signs. (−3) × 4 is −12. Use brackets.
• Forgetting to divide by the magnitudes. cos θ = a · b ÷ (|a||b|), not just a · b.
• Calculator in radians. cos⁻¹ gives radians unless your calculator is in degree mode.

Practise vectors in a short timed paper

Dot products come up in College Linear Algebra and AP Physics C: Mechanics practice, and in A-level Maths: Mechanics and A-level Further Maths in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

Read next

Maths · Vectors and matricesAngle between two vectors calculator: 2D and 3DFind the angle between two vectors in 2D or 3D from the dot product, cos θ = a·b ÷ (|a||b|), with the magnitudes, worked examples and a quick check.