벡터와 극좌표: 점을 표현하는 두 가지 방법

Vector operations, dot products, and how polar coordinates convert to rectangular — the building blocks of physics and engineering math.

7 분 TEKS 5D-5I Pre-Calculus

Vectors in component form

A vector v = ⟨a, b⟩ has magnitude |v| = √(a² + b²) and direction θ = arctan(b/a) (adjusted for quadrant).

Operations

Addition: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩
Scalar multiplication: k·⟨a, b⟩ = ⟨ka, kb⟩
Dot product: ⟨a, b⟩ · ⟨c, d⟩ = ac + bd (SCALAR result)

What the dot product tells you

u · v = |u| |v| cos θ, where θ is the angle between them. So:

u · v = 0 ⟺ u ⊥ v (orthogonal)
u · v > 0 → acute angle; u · v < 0 → obtuse
cos θ = (u · v)/(|u| |v|) gives the angle

Polar coordinates

A point can be located by (r, θ) where r is distance from origin and θ is angle from positive x-axis. Convert to rectangular:

x = r cos θ
y = r sin θ
r² = x² + y²
tan θ = y/x

Common polar curves

r = constant → circle of that radius centered at origin
θ = constant → line through origin at that angle
r = a sin θ or r = a cos θ → circle of diameter a, passing through origin
r = a ± a cos θ → cardioid (heart shape)
r = a sin(nθ) → rose with n petals (odd n) or 2n petals (even n)

Check yourself

📌 Convert

The polar point (r, θ) = (4, π/2) has rectangular coordinates:

Practice with real CBE questions

Pre-Calc Sem B practice for vectors and polar.