Linear Algebra
Class 12

Vector Dot Product

ab=abcosθ=a1b1+a2b2+a3b3\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3
00.511.522.53−20−1001020
a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃

Adjust Variables

Vector magnitude
mag =
110

Scalar product measures alignment of vectors. Perpendicular vectors have dot product = 0.

Real-World Applications

Physics — Work done: W = F · d = |F||d|cos(θ).

Machine learning — Cosine similarity for document comparison.

Computer graphics — Lighting calculations (Lambert's law).

Recommendation systems — User-item similarity.

Signal processing — Correlation between signals.

NLP — Word embeddings similarity.

Robotics — Force projection calculations.

Game physics — Collision response calculations.

Navigation — GPS position fixing.

Finance — Portfolio beta calculation.

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