Vectors and Parametric Equations Flashcards

Grade 11 · Math · Flashcards

Essential vector operations, properties, and parametric equation concepts.

Flashcards

Vector in 2D: v = ⟨a, b⟩; represents displacement a in x-direction and b in y-direction
Vector magnitude: |v| = √(a² + b²) for v = ⟨a, b⟩
Dot product: u · v = a₁a₂ + b₁b₂; scalar result; equals |u||v|cos(θ)
Perpendicular vectors: u ⊥ v if and only if u · v = 0
Parallel vectors: u ∥ v if v = ku for some scalar k (one is a scalar multiple of the other)
Unit vector: v̂ = v/|v|; has magnitude 1, same direction as v
i and j unit vectors: î = ⟨1, 0⟩ (x-direction), ĵ = ⟨0, 1⟩ (y-direction); any vector = aî + bĵ
Angle between vectors: cos(θ) = (u · v) / (|u| · |v|)
Parametric equations: Express x and y as separate functions of a parameter t: x = f(t), y = g(t)
Eliminating the parameter: Solve one parametric equation for t, substitute into the other to get y = f(x)
Parametric circle: x = r·cos(t), y = r·sin(t); traces a circle of radius r counterclockwise
Parametric line equation: Through point P_0 = (x_0, y_0) with direction d = ⟨a, b⟩: x = x_0 + at, y = y_0 + bt
Vector projection: proj_v u = (u · v / |v|²) v; component of u in the direction of v