3D Vectors & Operations

3D vectors are tuples (x, y, z).

Operations:

• Add: a + b = (ax+bx, ay+by, az+bz).
• Sub: a - b similarly.
• Scalar mul: a * s = (axs, ays, az*s).
• Length: |a| = √(ax² + ay² + az²).
• Normalize: a / |a| → unit vector.
• Dot: a · b = axbx + ayby + az*bz.
• Cross: a × b = (aybz - azby, azbx - axbz, axby - aybx).

Cross product:

• Result perpendicular to both inputs.
• Right-hand rule for direction.
• |a × b| = |a| * |b| * sin(angle).

Used heavily in graphics:

• Surface normal: n = (b - a) × (c - a) for triangle abc.
• Tangent space: cross of UV partial derivatives.
• Camera basis: cross to construct orthonormal frame.

Normalize:

Distance between points: |a - b|.

Reflection: r = v - 2*(v·n)*n.

Refraction (Snell's law): more complex; uses indices of refraction.

Coordinate systems:

• Right-handed: +X right, +Y up, +Z out of screen.
• Left-handed: +Z into screen.
• DirectX is left-handed; OpenGL is right-handed.
• Conversion: flip Z.

NumPy or GLM library for production. Hand-coded for educational toys.