Dot Product Calculator

What the dot product is

The dot product (also called the scalar product) is one of the most basic operations in linear algebra. It takes two vectors — ordered lists of numbers of the same length — and combines them into a single number. Unlike adding two vectors (which gives you another vector), the dot product always gives you a scalar.

The formula: a · b = Σ aᵢbᵢ

For two vectors with $n$ components each, $a = (a_1, a_2, \dots, a_n)$ and $b = (b_1, b_2, \dots, b_n)$, the dot product is:

$a \cdot b = a_1b_1 + a_2b_2 + \dots + a_nb_n = \sum_{i=1}^{n} a_i b_i$

In words: multiply each pair of matching components, then add all of those products together.

Step by step

Take $a = (1, 2, 3)$ and $b = (4, 5, 6)$:

1. Multiply the first components: $1 \times 4 = 4$.
2. Multiply the second components: $2 \times 5 = 10$.
3. Multiply the third components: $3 \times 6 = 18$.
4. Add the products: $4 + 10 + 18 = 32$.

So $(1,2,3) \cdot (4,5,6) = 32$. The calculator above walks through this same table-then-sum breakdown for whatever numbers you enter.

Component count must match

Because every term in the sum needs a partner, both vectors must have the same number of components. A 3-number vector can only be dotted with another 3-number vector; pairing it with a 2-number or 4-number vector leaves a term without a match, so the operation is undefined. This calculator flags mismatched lengths instead of guessing at a partial answer.

What the result tells you

The dot product is also the building block for vector magnitude (length): dotting a vector with itself and taking the square root gives its length, $\lVert a \rVert = \sqrt{a \cdot a}$ — the same square-root operation behind the Square Root calculator, and closely related to the "always non-negative" magnitude idea in the Absolute Value calculator.

Worked examples

Example 1 — three components: $(1,2,3) \cdot (4,5,6) = 1(4) + 2(5) + 3(6) = 4 + 10 + 18 = 32$.

Example 2 — a negative component: $(2,-1) \cdot (3,4) = 2(3) + (-1)(4) = 6 - 4 = 2$.

Example 3 — decimals stay exact: $(1.5, 2) \cdot (2, 2) = 1.5(2) + 2(2) = 3 + 4 = 7$.

Example 4 — perpendicular vectors: $(1,0) \cdot (0,1) = 1(0) + 0(1) = 0$, confirming the two vectors are orthogonal.

Where the dot product shows up

The dot product is a gateway idea into linear algebra: it underlies vector length and distance, the angle between two directions, projections (how much of one vector lies along another), and — in higher dimensions — the "similarity" scores used throughout machine learning and search. Physics uses it to compute work (force dotted with displacement); computer graphics uses it for lighting calculations; data science uses it for cosine similarity between feature vectors. Once you're comfortable multiplying matching components and summing the results, you have the core building block for all of these.