Dot Product Calculator

The two formulas

Component form for vectors a = (a1, a2, ..., an) and b = (b1, b2, ..., bn):

a . b = a1*b1 + a2*b2 + ... + an*bn

Geometric form:

a . b = |a| * |b| * cos(theta)

The two expressions are equal; setting them equal gives you cos(theta) = (a.b) / (|a| |b|), the cosine similarity formula.

What the sign tells you

Worked example

a = (3, 4), b = (1, 2):

• Components: 31 + 42 = 3 + 8 = 11
• |a| = sqrt(9 + 16) = 5
• |b| = sqrt(1 + 4) = sqrt(5) ~ 2.236
• cos(theta) = 11 / (5 * 2.236) = 0.9839
• theta = arccos(0.9839) ~ 10.3 degrees

The vectors point almost the same way, hence the large positive dot product.

Common applications

• Cosine similarity for recommender systems: how similar are two embedding vectors?
• Projection of one vector onto another: proj_b(a) = (a . b) / |b|^2 * b.
• Work in physics: W = F . d, force dotted with displacement.
• Graphics lighting: surface normal dotted with light direction gives diffuse brightness (Lambert’s law).
• Orthogonality check: quickly verify whether two basis vectors are perpendicular.

Tips

• Normalize before comparing. If you only care about direction (similarity), divide each vector by its magnitude first; then the dot product equals the cosine directly.
• Watch the dimension. A 3-vector dotted with a 4-vector is undefined; pad or truncate deliberately, not accidentally.
• Numerical stability. For very large vectors, floating-point accumulation can lose precision; use double precision or Kahan summation when dimension exceeds a few thousand.