Game Vector Math Primitives

Use this skill when game code depends on vector operations whose geometric meaning matters. Prefer clear primitives and named intent over opaque formula paste.

Primary source: Geometry for Programmers by Oleksandr Kaleniuk (https://www.manning.com/books/geometry-for-programmers), transformed and paraphrased, especially chapters 2 and 9.

Core Workflow

1. Identify the geometric question: angle, projection, side, area, normal,

volume, distance, facing, or interpolation.

1. Choose the primitive that answers that question directly.
2. Check the input assumptions: dimensionality, coordinate space, units,

handedness, normalization, and zero-length vectors.

1. Preserve sign when orientation or sidedness matters. Discard sign only when

the caller truly needs magnitude.

1. Name intermediate values by meaning: normal, signed_area2,

light_factor, plane_distance, facing, projection_t.

1. Add tests that exercise orientation, reversed winding, orthogonal vectors,

parallel vectors, and zero-length inputs.

Primitive Guide

Dot Product

Use dot products for:

• Projection onto a direction.
• Angle/facing tests.
• Lambert-style lighting coefficients.
• Distance along a ray or segment.
• Checking orthogonality: dot equals zero under the numeric policy.

If both vectors are normalized, the dot product is the cosine of the angle between them. Clamp before inverse cosine if a display angle is needed.

Cross Product And 2D Cross Scalar

Use cross products for:

• Surface normals from two triangle edges.
• Signed 2D orientation tests.
• Twice the signed area of a 2D triangle.
• Detecting parallel vectors through near-zero magnitude.
• Building tangent frames with a known handedness.

For 2D, use the scalar a.x * b.y - a.y * b.x instead of constructing fake 3D vectors unless the codebase already uses a 3D type.

Triple Product

Use the scalar triple product for:

• Signed volume.
• Point side relative to an oriented plane.
• Point-to-plane distance when divided by the plane normal length.
• Tetrahedron orientation and containment-style predicates.

Keep the sign when side matters. Take absolute value only for unsigned volume or distance.

Guardrails

• Never normalize before checking vector length.
• Distinguish a direction vector from a point vector in naming and APIs.
• Keep normals in the same space as the vectors they are compared with.
• Recompute or transform normals correctly after nonuniform scale.
• Decide whether clockwise or counterclockwise winding is canonical.
• Use squared length when comparing distances and no square root is needed.
• Avoid inverse trigonometric functions in hot paths unless a real angle is

required. Dot thresholds are usually cheaper.

Common Mistakes

• Treating every vector as normalized because examples used unit vectors.
• Losing useful sign information by taking absolute values too early.
• Mixing coordinate spaces in one dot or cross operation.
• Using angle calculations where a dot-product threshold would be simpler.
• Flipping triangle winding without updating normals, culling, or orientation

predicates.