#L02 · Linear Algebra & Manifolds · Lv.2View in 3D Mind-Map

Dot Product & Cosine Similarity

J. W. Gibbs & Oliver Heaviside · 1881 CE

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

\text{sim}(\mathbf{a}, \mathbf{b}) = \cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}

Equates simple component-wise multiplication ∑ a_i b_i with geometric angle alignment ‖a‖‖b‖cos θ—powering LLM RAG vector search, Transformer Attention, and 3D lighting shaders.

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Vec a_x

Vec a_y

Vec b_x

Vec b_y

\mathbf{a}\cdot\mathbf{b} = (4)(2) + (2)(4) = 16.0, \quad \cos\theta = 0.800

내적 a · b

16.000

코사인 유사도 cos θ

0.8000

두 벡터 사잇각 θ

36.9°

Prerequisite Path (10 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#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#G01Pythagorean Theorem

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

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

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

STEP 7#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 8#G022D/3D Euclidean Distance Formula

d(P

STEP 9#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 10#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

🚀 Next Unlocks (Derived Formulas) (5)

#L033D Cross Product & Surface Normal Vector #L04Matrix Multiplication & Linear Transformation #L09Singular Value Decomposition (SVD) & PCA #L10Quaternions & 3D Spherical Interpolation (SLERP)#P08Pearson Correlation & Linear Regression (OLS)

Geometric & Intuitive Insight

두 벡터가 같은 방향(0°)을 바라보면 cos θ = 1 (유사도 최대), 수직(90°)이면 내적이 0 (무관함/직교), 반대 방향(180°)이면 -1 이 됩니다. 게임에서 적이 내 시야각 안에 있는지, 태양빛이 표면에 얼마나 수직으로 비치는지 단 한 줄로 판별합니다!

Step-by-Step Derivation

Step 1.1단계: 두 벡터 a, b와 차이 벡터 (a - b)가 이루는 삼각형에 제2코사인법칙을 적용합니다 → ‖a - b‖² = ‖a‖² + ‖b‖² - 2‖a‖‖b‖cos θ
Step 2.2단계: 좌변을 성분별로 전개하면 ∑(a_i - b_i)² = ∑a_i² + ∑b_i² - 2∑a_i b_i 입니다.
Step 3.3단계: 양변의 ‖a‖² + ‖b‖² 을 소거하고 -2로 나누면 ∑ a_i b_i = ‖a‖‖b‖cos θ 가 완벽히 증명됩니다.

AI & Coding Real-World Use

LLM Transformer Attention (Q·K^T), RAG vector DB cosine search, and 3D Lambertian lighting (N·L)

#LLM Self-Attention Q·K#RAG Vector DB#3D Lighting Shader

const cosSim = (a:number[], b:number[]) => dot(a,b) / (norm(a) * norm(b));

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