Derived Vector Operations

By Victor Arce

Once you have the dot product and subtraction, a surprising amount of geometry falls out for free. Angle, distance, reflection and smooth interpolation aren't new machinery — they're those two operations wearing different hats.

Step through each on the grid below, using a = (3, 1) and b = (1, 2).

Angle — the dot product, normalized

The dot product already measures "how much two vectors point the same way." Divide it by both lengths and you get exactly the cosine of the angle between them:

cos θ = (a · b) / (|a| · |b|)

Here that's 5 / (√10·√5) ≈ 0.707, so θ ≈ 45°. If the dot product is zero the vectors are perpendicular; if it's negative they point more than 90° apart.

Distance — the length of a difference

The distance between the two points is just the magnitude of a − b:

dist(a, b) = |a − b| = √((3−1)² + (1−2)²) = √5 ≈ 2.24

Subtraction gives the vector from one point to the other; its length is the distance.

Reflection — project, then go twice as far

To mirror b across the line through a, first project b onto a (the shadow it casts), then continue the same distance past it:

b′ = 2·proj_a(b) − b = (2, −1)

The two dashed drops in the animation are mirror images across a's line — that's what makes it a reflection. (Projection itself comes from the core operations article.)

Lerp — blending one vector into another

Linear interpolation walks in a straight line from a to b:

lerp(a, b, t) = a + t·(b − a)

At t = 0 you're at a, at t = 1 you're at b, and t = 0.5 is the midpoint (2, 1.5). Sweep t from 0 to 1 and you get smooth motion between the two — the move behind almost every animation and transition.

Back to Vector Math