Stewart Calculus · 8th Edition
Chapter 12: Vectors & Geometry of Space
Every key formula from Stewart Calculus Chapter 12, in one searchable page. Click a card to study it — worked examples included.
12.1–12.2 · COORDS & VECTORS
dist = √((Δx)² + (Δy)² + (Δz)²)
Sphere: (x−h)² + (y−k)² + (z−l)² = r²
|a| = √(a₁² + a₂² + a₃²), û = a/|a|
i, j, k form
a = ⟨a₁,a₂,a₃⟩ = a₁i + a₂j + a₃k
c·a = ⟨ca₁, ca₂, ca₃⟩ — scale component-wise.
Displacement
PQ = ⟨q₁−p₁, q₂−p₂, q₃−p₃⟩
Head minus tail: the vector from P to Q.
WORKED EXAMPLE
a = ⟨2,−2,1⟩: |a| = √(4+4+1) = 3;
unit vector ⟨2/3, −2/3, 1/3⟩.
unit vector ⟨2/3, −2/3, 1/3⟩.
Zero vector 0 = ⟨0,0,0⟩: |0| = 0, no direction.
|a + b| ≤ |a| + |b| (triangle inequality).
12.3 · DOT PRODUCT
a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ
Orthogonal ⟺ a · b = 0.
Projection
projab = a · b|a|²a, scalar comp = a · b|a|
WORKED EXAMPLE
a = ⟨1,2,3⟩, b = ⟨−1,0,2⟩:
a · b = 5; cos θ = 5/√70.
a · b = 5; cos θ = 5/√70.
Direction angles
cos α = a₁/|a|, cos²α + cos²β + cos²γ = 1
WORKED EXAMPLE
a = ⟨1,1,0⟩, b = ⟨2,0,1⟩:
projab = ⟨1,1,0⟩, scalar comp. √2.
projab = ⟨1,1,0⟩, scalar comp. √2.
12.4 · CROSS PRODUCT
a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|
|a × b| = |a||b|sin θ = parallelogram area
a × b ⊥ both; a × b = −(b × a).
WORKED EXAMPLE
Same a, b: a × b = ⟨4, −5, 2⟩.
Check: a · ⟨4,−5,2⟩ = 0 ✓
Check: a · ⟨4,−5,2⟩ = 0 ✓
Scalar triple product
a · (b × c) = determinant = parallelepiped volume
= 0 ⟺ a, b, c are coplanar.
WORKED EXAMPLE
Triangle P(0,0,0), Q(1,2,0), R(2,1,0):
PQ × PR = ⟨0,0,−3⟩ ⇒ area 3/2.
PQ × PR = ⟨0,0,−3⟩ ⇒ area 3/2.
12.5 · LINES & PLANES
Line
r = r₀ + tv; x − x₀a = y − y₀b = z − z₀c
Plane (normal ⟨a,b,c⟩)
a(x − x₀) + b(y − y₀) + c(z − z₀) = 0
Point→plane: |ax₀+by₀+cz₀+d|√(a²+b²+c²).
WORKED EXAMPLE
Through (1, 0, −1), normal ⟨2,−1,3⟩:
2x − y + 3z + 1 = 0.
2x − y + 3z + 1 = 0.
Two planes
Parallel ⟺ n₁ ∥ n₂; cos θ = |n₁·n₂|/(|n₁||n₂|)
Skew lines: neither parallel nor intersecting.
Point → line distance
d = |v × PQ||v| (Q on line, direction v)
WORKED EXAMPLE
Through (2,1,−3), direction ⟨1,−2,4⟩:
(x−2)/1 = (y−1)/−2 = (z+3)/4.
(x−2)/1 = (y−1)/−2 = (z+3)/4.
12.6 · QUADRIC SURFACES
Ellipsoid: x²/a² + y²/b² + z²/c² = 1
Cone: z² = x² + y² · Paraboloid: z = x² + y²
Hyperboloids: one sheet −1, two sheets +1 on RHS
Cone: z² = x² + y² · Paraboloid: z = x² + y²
Hyperboloids: one sheet −1, two sheets +1 on RHS
KEY NOTES
- Traces (cross-sections) reveal the shape.
Saddle
Hyperbolic paraboloid: z = x² − y²
Cylinders
Missing a variable ⇒ extrudes along that axis
x² + y² = 1: circular cylinder along z.
WORKED EXAMPLE
4x² + y² − z² = 4 ⇒
x² + y²/4 − z²/4 = 1: hyperboloid of one sheet.
x² + y²/4 − z²/4 = 1: hyperboloid of one sheet.
WATCH OUT!
Mistakes that cost points
- a × b = −(b × a) — order matters!
- Dot ⇒ scalar; cross ⇒ vector
- Symmetric equations break if a direction # is 0
- Plane distance needs ax+by+cz+d = 0 form
- |a × b| is an AREA, not a volume
- Check angles with cos²α+cos²β+cos²γ = 1
- Triple product: cyclic order keeps the sign, a swap flips it
- Symmetric line equations need every direction number ≠ 0
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