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vector-algebra

Abstract math for vector algebra in 2D and 3D Euclidean space. USE FOR: implementing vector operations (add, subtract, negate, scalar multiply/divide); dot product and geometric interpretation; 3D cross product; 2D pseudo-cross product (scalar z-component); vector magnitude and normalization; angle between two vectors; linear interpolation; checking zero/unit/finite vectors. 2D is a special case of 3D with z=0. DO NOT USE FOR: quaternion operations (use quaternion-algebra); rotation representations (use 3d-rotation-theory); coordinate system conversions (use coordinate-system-conversions).

中文简介正在准备中。当前显示原文说明。

NetFabric/NetFabric.Numerics星标 36(GitHub 36 + cccskills 0)

实测评分 未实测

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包含的文件（3）

  • SKILL.md2.5 KB
  • references/formulas.md2.0 KB
  • references/numerical-stability.md2.3 KB

SKILL.md（原文）

安装前可以查看交给 AI 代理的指令内容。

Vector Algebra

Point − Point = Vector (displacement). Point + Vector = Point (translation).

Operations (component-wise)

OperationFormulaResult type
Additiona + b = (aₓ+bₓ, aᵧ+bᵧ[, a_z+b_z])Vector
Subtractiona − bVector
Scalar multiplykv = (kvₓ, kvᵧ[, kv_z])Vector
Scalar dividev/kVector
Negation−v = (−vₓ, −vᵧ[, −v_z])Vector

Magnitude & Normalization

‖v‖  = √(vₓ² + vᵧ² [+ v_z²])        (Euclidean norm)
‖v‖² = vₓ² + vᵧ² [+ v_z²]           (avoid sqrt when comparing)
v̂    = v / ‖v‖                         (unit vector; undefined if ‖v‖ = 0)

Dot Product

a · b = aₓbₓ + aᵧbᵧ [+ a_z b_z] = ‖a‖ ‖b‖ cos θ

a·b = 0 → perpendicular; a·b = ‖a‖‖b‖ → parallel; sign → acute/obtuse.

Cross Product (3D only)

a × b = (aᵧb_z − a_z bᵧ,  a_z bₓ − aₓb_z,  aₓbᵧ − aᵧbₓ)
‖a × b‖ = ‖a‖ ‖b‖ sin θ

Direction: right-hand rule. Anti-commutative: a × b = −(b × a)

2D pseudo-cross (scalar z-component only): aₓbᵧ − aᵧbₓ — positive = CCW, negative = CW.

Angle Between Vectors

θ = arccos(clamp(a·b / (‖a‖‖b‖), −1, 1)) result ∈ [0, π]

Special Vectors

NameCondition
Zero‖v‖ = 0
Unit (normalized)‖v‖ = 1
Basis 2DUnitX=(1,0), UnitY=(0,1)
Basis 3DUnitX=(1,0,0), UnitY=(0,1,0), UnitZ=(0,0,1)

Reference Files

FileLoad When
references/formulas.mdProjection, triple product, distance formulas, point operations
references/numerical-stability.mdNear-zero normalization, dot cancellation, magnitude comparison

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