Free Dot Product Calculator

This dot product calculator finds the dot product of two or more vectors and shows the working, step by step. Enter each vector with parentheses, brackets, angle brackets, or just one per line — whatever's easiest.

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Dot Product Calculator

Define each vector with parentheses ( ), square brackets [ ], angle brackets < >, or a new line. Separate the terms inside each vector with a comma. Every vector must have the same number of terms.

Input Rules

  • Enter 2 or more vectors, each with the same number of terms.
  • Vectors can hold integers and decimals (including negatives), but not fractions, functions, or variables.
  • Accepted vector wrappers: (1,2,3), [1,2,3], <1,2,3>, or just 1,2,3 on its own line.

Answer

Solution:

Share This Calculation

Opening this link fills in your vectors and shows this result automatically.

Enter your vectors and the dot product, worked out step by step, will appear here.

What Is a Dot Product?

A dot product takes two vectors of matching length and collapses them down into a single number. You multiply each pair of matching entries together, then add all of those products up. It shows up constantly in physics (work, projections) and in computing (similarity between two lists of numbers).

The Formula

For vectors a = <a₁, a₂, a₃...aₙ> and b = <b₁, b₂, b₃...bₙ>:

a · b = (a₁ × b₁) + (a₂ × b₂) + (a₃ × b₃) + ... + (aₙ × bₙ)

The same idea extends to three or more vectors — multiply the matching entries from every vector together at each position, then add those products up.

Worked Example

Find a · b when a = <3, 5, 8> and b = <2, 7, 1>:

a · b = (3 × 2) + (5 × 7) + (8 × 1)
a · b = 6 + 35 + 8
a · b = 49

A Quick Geometric Way to Think About It

Beyond the arithmetic, the dot product also tells you something about direction. A positive dot product means two vectors generally point the same way, a negative one means they point roughly opposite directions, and a dot product of exactly zero means they're perpendicular to each other.

Using the Share, Print and Copy Options

  • Share: After you calculate, a ready-made link appears with your exact vectors baked in. Send it to anyone and their page opens already filled in with your result showing.
  • Print: Prints just your vectors and the worked solution — no ads, menus or surrounding page clutter.
  • Copy: Copies a plain-text version of the answer straight to your clipboard, ready to paste anywhere.

Frequently Asked Questions (FAQs)

Yes. Enter as many vectors as you like, each with the same number of terms, and this calculator multiplies the matching entries across all of them at each position before adding the products together.

The dot product works by pairing up matching positions between vectors. If one vector had an extra term with nothing to pair it against, the operation simply isn't defined, so every vector needs the same number of terms.

For two vectors, a dot product of exactly zero means they're perpendicular (at a 90-degree angle) to each other. It's a quick way to test for perpendicularity without measuring any angles directly.

Yes, parentheses, square brackets and angle brackets can all be mixed freely in the same input, or skipped entirely by putting one vector per line instead.

No. The link only encodes the exact vectors you typed in, so anyone opening it sees the same calculation you did, nothing more.

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