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Grade 12 Math Lab: Analysis & Inference FREE
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Vector Projections
Learning Objectives
- Compute dot products
- Project vectors
- Find components along directions
Key Concepts
Dot Product
a⃗·b⃗ = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃
Projection
proj_b(a) = (a⃗·b⃗/|b⃗|²)b⃗
Theory
**Dot Product:** a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃
**Scalar projection** of a⃗ onto b⃗:
comp_b(a) = a⃗·b⃗ / |b⃗|
**Vector projection:**
proj_b(a) = (a⃗·b⃗ / |b⃗|²) b⃗
**Applications:** Work = F⃗·d⃗, shadow length, decomposing forces.
**Orthogonal:** a⃗·b⃗ = 0 ⟺ a⃗ ⊥ b⃗
Examples
Project ⟨3,4⟩ onto ⟨1,0⟩
Solution: proj = (3·1+4·0)/(1) · ⟨1,0⟩ = ⟨3,0⟩