Calculus III Exam 1 Formula Sheet (Geometry of Space and Vector Functions)

MAC2313: Calculus 3 Formula Sheet
Lectures 1–9: Vectors, Geometry, and Motion


Vectors & Dot Products Quadric Surfaces

x2 y2 z2
• Magnitude: ∥v∥ = • Ellipsoid:
p
v12 + v22 + v32 a2 + b2 + c2 =1
x2 y2
• Unit Vector: u = v • Elliptic Paraboloid: z
c = a2 + b2
∥v∥
x2 y2
• Hyperbolic Paraboloid: z
= −
• Dot Product: a · b = a1 b1 + a2 b2 + a3 b3 = c a2 b2

∥a∥∥b∥ cos θ 2 2 2
• Cone: zc2 = xa2 + yb2

• Orthogonality: a ⊥ b ⇐⇒ a · b = 0 x2 y2 z2
• Hyperboloid (1 Sheet): a2 + b2 − c2 =1
2 y2 z2
• Scalar Projection (comp): compa b = a·b
∥a∥
• Hyperboloid (2 Sheets): − xa2 − b2 + c2 =1

 
• Vector Projection (proj): proja b = a·b
∥a∥2 a Vector Functions (Space Curves)

• Work: W = F · D
Rb
• Arc Length: L = a
∥r′ (t)∥ dt
r′ (t)
• Unit Tangent: T(t) = ∥r′ (t)∥
The Cross Product T′ (t)
• Unit Normal: N(t) = ∥T′ (t)∥

i j k • Binormal: B(t) = T(t) × N(t)
• Formula: a × b = a1 a2 a3
∥T′ (t)∥ ∥r′ (t)×r′′ (t)∥
b1 b2 b3 • Curvature (κ): κ = ∥r′ (t)∥ = ∥r′ (t)∥3


• Magnitude: ∥a × b∥ = ∥a∥∥b∥ sin θ (Area of par-
allelogram) Motion in Space

• Parallel: a ∥ b ⇐⇒ a × b = 0 • Velocity: v(t) = r′ (t)
• Speed: ∥v(t)∥
• Volume (Triple Scalar): V = |a · (b × c)|
• Acceleration: a(t) = r′′ (t) = aT T + aN N
r′ (t)·r′′ (t)
Lines & Planes • Tangential Comp (aT ): ∥r′ (t)∥

∥r′ (t)×r′′ (t)∥
• Normal Comp (aN ): ∥r′ (t)∥
• Line Equation: r = r0 + tv

– Parametric: x = x0 + at, y = y0 + bt, z =
z0 + ct
x−x0 y−y0 z−z0
– Symmetric: a = b = c


• Plane Equation: a(x−x0 )+b(y−y0 )+c(z−z0 ) =
0 or n · (r − r0 ) = 0

|ax1 +by1 +cz1 +d|
• Distance (Point to Plane): D = √
a2 +b2 +c2


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