#L01 · Linear Algebra & Manifolds · Lv.1View in 3D Mind-Map

Vector Operations, L1/L2 Norms & Normalization

Hermann Grassmann & J. W. Gibbs · 1844 CE

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2},\quad \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|_2}

\|\mathbf{v}\|_1 = \sum_{i=1}^{n} |v_i|

Measures vector magnitude via L1/L2 norms and scales vectors to unit length v̂ = v/‖v‖—essential for 3D game movement direction and L1/L2 regularization in machine learning.

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\|\mathbf{v}\|_2 = \sqrt{3^2+4^2+0^2} = 5.000, \quad \hat{\mathbf{v}} = \left(0.60, 0.80, 0.00\right)

유클리드 노름 ||v||₂

5.0000

단위벡터 v̂ (x, y, z)

(0.60, 0.80, 0.00)

맨해튼 노름 ||v||₁

7.00

Prerequisite Path (6 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 5#G022D/3D Euclidean Distance Formula

d(P

STEP 6#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

🚀 Next Unlocks (Derived Formulas) (3)

#C08Partial Derivatives & Gradient Vector (∇f)#L02Dot Product & Cosine Similarity #L033D Cross Product & Surface Normal Vector

Geometric & Intuitive Insight

게임에서 위쪽 키와 오른쪽 키를 동시에 누르면 이동 벡터가 (1, 1)이 되어 속도가 √2 ≈ 1.414배 빨라지는 버그가 생깁니다. 이를 L2 노름 ‖v‖로 나누어 정규화(v̂)하면 어느 방향이든 속도가 1로 일정해집니다!

Step-by-Step Derivation

Step 1.1단계: n차원 벡터 v = (v₁, v₂, …, v_n) 의 원점 거리는 피타고라스 정리에 의해 ‖v‖₂ = √(v₁² + v₂² + … + v_n²)
Step 2.2단계: 각 성분을 ‖v‖₂ 로 나눈 벡터 v̂ = v / ‖v‖₂ 의 길이를 재면 ‖v̂‖₂ = ‖v‖₂ / ‖v‖₂ = 1 이 됩니다.
Step 3.3단계: 절댓값 합 ‖v‖₁ = ∑|v_i| 는 격자 도로를 따라 이동하는 맨해튼 거리(L1 Norm)를 나타냅니다.

AI & Coding Real-World Use

Deep learning L1/L2 Weight Decay regularization, LayerNorm, and 3D game movement vector normalization

#L1/L2 Regularization#3D Vector Normalize#Embedding Norm

const norm2 = Math.hypot(vx, vy, vz); const unit = [vx/norm2, vy/norm2, vz/norm2];

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