#L03 · Linear Algebra & Manifolds · Lv.2View in 3D Mind-Map

3D Cross Product & Surface Normal Vector

William Rowan Hamilton & J. W. Gibbs · 1843 CE

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y,\; a_z b_x - a_x b_z,\; a_x b_y - a_y b_x)

\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\|\|\mathbf{b}\|\sin\theta,\quad (\mathbf{a}\times\mathbf{b}) \perp \mathbf{a}, \mathbf{b}

Produces a 3D vector perpendicular to both input vectors (right-hand rule) whose magnitude equals the area of the parallelogram they span.

🧮 Live Formula Calculator & Plotter

Vec a_x

Vec a_y

Vec b_x

Vec b_y

\mathbf{a}\times\mathbf{b} = (0,\; 0,\; 2\cdot3 - 0\cdot0) = (0,\; 0,\; 6.0)

법선 벡터 a×b (z성분)

(0, 0, 6.00)

평행사변형 넓이 ||a×b||

6.000

삼각형 폴리곤 넓이 (½)

3.000

Prerequisite Path (11 STEP)

Full Path Analyzer

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#G03Pi, Radian Measure & Circular Sector

l = r\theta

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

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

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G05Law of Sines & Law of Cosines

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

STEP 9#G04Unit Circle Trigonometry & Identities

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

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

STEP 11#L033D Cross Product & Surface Normal Vector

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y

🚀 Next Unlocks (Derived Formulas) (2)

#L05Determinant & Matrix Inverse #L10Quaternions & 3D Spherical Interpolation (SLERP)

Geometric & Intuitive Insight

오른손 네 손가락을 벡터 a에서 b로 감아쥐었을 때 엄지손가락이 가리키는 수직 방향이 바로 a × b 입니다. 3D 모델의 세 꼭짓점으로 만든 두 변을 외적하면 빛을 반사할 표면의 방향(법선벡터)이 즉시 구해집니다.

Step-by-Step Derivation

Step 1.1단계: 미지의 벡터 n = (x, y, z) 가 a·n = 0 및 b·n = 0 (두 벡터 모두와 수직)을 만족하도록 연립방정식을 세웁니다.
Step 2.2단계: 행렬식 기호를 사용해 i(a_y b_z - a_z b_y) - j(a_x b_z - a_z b_x) + k(a_x b_y - a_y b_x) 를 도출합니다.
Step 3.3단계: 라그랑주 항등식 ‖a×b‖² + (a·b)² = ‖a‖²‖b‖² 에 cos²θ + sin²θ = 1 을 대입하면 ‖a×b‖ = ‖a‖‖b‖sin θ

AI & Coding Real-World Use

3D mesh triangle surface normals, back-face culling, and physics torque (τ = r × F) calculation

#3D Surface Normal#Physics Torque r×F#Back-face Culling

const cross = (a:number[], b:number[]) => [a[1]*b[2]-a[2]*b[1], a[2]*b[0]-a[0]*b[2], a[0]*b[1]-a[1]*b[0]];

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