Calculating the Dot Product and the Angle

The dot product turns two vectors into a single number. That number, combined with the two lengths, is enough to recover the exact angle between them.

Theorem How to compute it

Multiply the x-components together, multiply the y-components together, then add.

The answer is a scalar, not a vector. This is the key difference from the cross product, which returns a vector.

Example Two dot products

x-parts:

y-parts:

⟹ 63

⟹ 5

Negative components are handled exactly as they look — , which then gets added.

Theorem The angle formula

Divide the dot product by the product of the two lengths, then take the inverse cosine.

where

Example Finding the angle

Find the angle between and .

Step 1 — the dot product:

Step 2 — the length of A:

Step 3 — the length of B:

Step 4 — apply the formula:

Step 5 — take the inverse cosine.

⟹ θ ≈ 14.3°

A small angle makes sense: the cosine came out close to 1, meaning the vectors nearly point the same way.

Note Two shortcuts worth knowing

, so

and θ = 90°.

⟹ a dot product of zero means the vectors are perpendicular

, with lengths 5 and 10, giving

⟹ θ = 0°, the vectors are parallel

Checking for a zero dot product is the quickest test for perpendicularity there is.

Summary

A · B = a₁b₁ + a₂b₂, and the answer is always a number.
⟨3, 4⟩ · ⟨5, 12⟩ = 63.
cos θ = (A · B) / (|A| |B|), then take the inverse cosine.
For those vectors cos θ = 63/65, giving θ ≈ 14.3°.
A dot product of 0 means perpendicular; cos θ = 1 means parallel.