Dot product

The Euclidean dot product multiplies corresponding coordinates and sums the products. It measures directional alignment together with magnitude.

Ask the Linear algebra assistant1 min read · Updated September 9, 2026

Definition

The Euclidean dot product multiplies corresponding coordinates and sums the products. It measures directional alignment together with magnitude.

A large dot product can result from long vectors, strong alignment, or both. Normalize first when you want only direction.

(1, 2) · (3, 4) = 1 × 3 + 2 × 4 = 11.

For real vectors, $x^Ty=\sum_i x_i y_i=\|x\|\|y\|\cos\theta$ .

Use dot products in linear models, projections, and similarity search, checking that dimensions and coordinate meanings match.

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