Dot and Cross Products

Dot and Cross Products

Two fundamental vector operations power almost all lighting math in the ray tracer.

Dot Product

The dot product returns a scalar — the sum of component-wise products:

dot(a, b) = a.x*b.x + a.y*b.y + a.z*b.z

double dot(Tuple a, Tuple b) {
    return a.x*b.x + a.y*b.y + a.z*b.z + a.w*b.w;
}

The dot product has a geometric meaning: dot(a, b) = |a| * |b| * cos(θ) where θ is the angle between them. For normalized vectors:

• dot = 1 → same direction
• dot = 0 → perpendicular
• dot = -1 → opposite directions

Cross Product

The cross product returns a vector perpendicular to both inputs:

cross(a, b).x = a.y*b.z - a.z*b.y
cross(a, b).y = a.z*b.x - a.x*b.z
cross(a, b).z = a.x*b.y - a.y*b.x

Tuple cross(Tuple a, Tuple b) {
    return Tuple(
        a.y*b.z - a.z*b.y,
        a.z*b.x - a.x*b.z,
        a.x*b.y - a.y*b.x,
        0
    );
}

Note: cross(a, b) = -cross(b, a). Order matters!

Your Task

Implement both dot and cross functions.

Expected output:

Dot: 20
Cross: -1 2 -1