Difficulty: Introductory-Intermediate | Prerequisites: MATH 124/125 (single-variable calculus), basic trigonometry
Chapter 12 is the foundation for everything in Calculus III. It extends the familiar 2D coordinate plane into three dimensions and introduces vectors, which become the language for describing motion, forces, and geometry throughout the rest of the course. If you are comfortable with 2D slopes, lines, and basic trig, you have what you need.
Chapter 12 sets up the coordinate system and algebraic tools you will use for the rest of Calculus III. You learn how to describe points, distances, and shapes in three dimensions, then how to manipulate vectors with the dot product (which measures alignment and gives projections) and the cross product (which produces a perpendicular vector and measures area). The chapter closes with equations for lines and planes in 3D and a catalogue of quadric surfaces.
ℝ³ (3D space)
The set of all ordered triples (x, y, z). Think of it as the full three-dimensional world, extending the flat ℝ² plane by adding a vertical axis.
First octant
The region where x ≥ 0, y ≥ 0, and z ≥ 0. In simple terms, this is the "positive corner" of 3D space, analogous to the first quadrant in 2D.
Magnitude (of a vector)
The length of a vector, written |v|. For v = ⟨v₁, v₂, v₃⟩, the magnitude is √(v₁² + v₂² + v₃²). Think of it as the distance from the origin to the tip of the vector.
Unit vector
A vector with magnitude 1. You get it by dividing any vector by its own magnitude: v / |v|. In simple terms, it keeps the direction but standardises the length to 1.
Dot product (scalar product)
a · b = a₁b₁ + a₂b₂ + a₃b₃. The result is a scalar (a number), not a vector. Equivalently, a · b = |a||b| cos θ where θ is the angle between them.
Orthogonal
Two vectors are orthogonal (perpendicular) when their dot product equals zero. Think of it as the vectors forming a 90-degree angle.
Component projection
comp_a(b) = (a · b) / |a| = |b| cos θ. This is a number that tells you how much of b lies in the direction of a.
Vector projection
proj_a(b) = ((a · b) / |a|²) a. This is a vector: the "shadow" of b onto the line of a.
Cross product (vector product)
a × b produces a vector that is perpendicular to both a and b. Its magnitude |a × b| = |a||b| sin θ equals the area of the parallelogram spanned by a and b. Think of it as the operation that finds a direction orthogonal to two given directions.
Direction vector
A vector v = ⟨a, b, c⟩ that is parallel to a line and defines which way the line runs.
Normal vector
A vector n = ⟨a, b, c⟩ that is perpendicular to a plane. It defines the plane's orientation in space.
Parametric equations (of a line)
x = x₀ + at, y = y₀ + bt, z = z₀ + ct. Each coordinate is expressed as a function of a single parameter t.
Quadric surface
A surface defined by a second-degree equation in x, y, and z. Think of it as the 3D analogue of conic sections (ellipses, parabolas, hyperbolas).
ℝ² is the set of all points in 2D; ℝ³ extends this with a third axis (z)
The first octant is the region where x ≥ 0, y ≥ 0, z ≥ 0
Distance between two points: √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Equation of a sphere with centre (h, k, l) and radius r: (x - h)² + (y - k)² + (z - l)² = r²
Coordinate planes and axes:
xy-plane: all points where z = 0
yz-plane: all points where x = 0
xz-plane: all points where y = 0
x-axis: y = 0 and z = 0
y-axis: x = 0 and z = 0
z-axis: x = 0 and y = 0
Magnitude: |v| = √(v₁² + v₂² + v₃²)
Vector from A(a₁, a₂, a₃) to B(b₁, b₂, b₃): AB = ⟨b₁ - a₁, b₂ - a₂, b₃ - a₃⟩
Unit vector: v / |v|
Trig components (2D): x = r cos θ, y = r sin θ, and x² + y² = r²
a · b = a₁b₁ + a₂b₂ + a₃b₃ (the result is always a scalar)
Geometric form: a · b = |a||b| cos θ
If a and b are orthogonal: a · b = 0
If a and b are parallel: a · b = |a||b| or a · b = -|a||b|
Component projection of b onto a: comp_a(b) = (a · b) / |a| = |b| cos θ
This is a signed number: positive when b points "with" a, negative when against
Vector projection of b onto a: proj_a(b) = ((a · b) / |a|²) a
This is a vector in the direction of a
a × b is computed via the determinant:
a × b = (a₂b₃ - a₃b₂) i - (a₁b₃ - a₃b₁) j + (a₁b₂ - a₂b₁) k
The result a × b is orthogonal to both a and b
Right-hand rule: curl fingers from a to b; thumb points in the direction of a × b
a × b = -b × a (order matters)
|a × b| = |a||b| sin θ = area of the parallelogram with sides a and b
If a and b are parallel: a × b = ⟨0, 0, 0⟩
A line needs a point r₀ = ⟨x₀, y₀, z₀⟩ and a direction vector v = ⟨a, b, c⟩
Vector form: ⟨x, y, z⟩ = ⟨x₀, y₀, z₀⟩ + t⟨a, b, c⟩
Parametric form: x = x₀ + at, y = y₀ + bt, z = z₀ + ct
Symmetric form: (x - x₀)/a = (y - y₀)/b = (z - z₀)/c
Classifying pairs of lines:
Parallel: direction vectors are scalar multiples of each other
Intersecting: they share a point (x, y, z) at some parameter values
Skew: they neither intersect nor are parallel (this cannot happen in 2D, only in 3D)
To find a line's equation: if you are given the info directly, use it. If you are given two points, compute the direction vector by subtracting one from the other.
A plane needs a point r₀ = ⟨x₀, y₀, z₀⟩ and a normal vector n = ⟨a, b, c⟩
Equation: a(x - x₀) + b(y - y₀) + c(z - z₀) = 0
Vector form: n · (r - r₀) = 0
Two planes are parallel when their normal vectors are parallel
Non-parallel planes intersect along a line
To find a plane's equation: if you have a point and a normal, plug in directly. If you have three points, form two vectors from those points and take their cross product to get the normal.
Line: ax + by = c
Ellipse: x²/a² + y²/b² = 1 (a circle when a = b)
Parabola: ax² + by = c or ax + by² = c
Hyperbola: x²/a² - y²/b² = 1
If one variable is entirely absent from the equation, the 2D shape is extended along the missing variable's axis. For example, x² + y² = 1 (missing z) is a circular cylinder running along the z-axis.
These are surfaces given by equations involving x, y, and z raised to at most the second power. To visualise them, use traces: set one variable to a constant and examine the resulting 2D cross-section.
The six standard quadric surfaces:
Ellipsoid: x²/a² + y²/b² + z²/c² = 1 (all traces are ellipses; a sphere when a = b = c)
Hyperboloid of one sheet: x²/a² + y²/b² - z²/c² = 1 (horizontal traces are ellipses, vertical traces are hyperbolas)
Hyperboloid of two sheets: x²/a² + y²/b² - z²/c² = -1 (two separate pieces)
Elliptic/circular cone: x²/a² + y²/b² - z²/c² = 0 (the surface passes through the origin)
Elliptic paraboloid: z = x²/a² + y²/b² (horizontal traces are ellipses, vertical traces are parabolas; opens upward)
Hyperbolic paraboloid (saddle): z = x²/a² - y²/b² (horizontal traces are hyperbolas, vertical traces are parabolas)
The variable raised to the first power (if one exists) determines the axis of symmetry. The sign pattern in the equation tells you which surface you have.
Topic | Formula |
|---|---|
Distance between two points | √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²) |
Sphere | (x - h)² + (y - k)² + (z - l)² = r² |
Magnitude of a vector | |
Dot product (algebraic) | a · b = a₁b₁ + a₂b₂ + a₃b₃ |
Dot product (geometric) | a · b = |
Component projection | comp_a(b) = (a · b) / |
Vector projection | proj_a(b) = ((a · b) / |
Cross product magnitude | |
Line (parametric) | x = x₀ + at, y = y₀ + bt, z = z₀ + ct |
Line (symmetric) | (x - x₀)/a = (y - y₀)/b = (z - z₀)/c |
Plane equation | a(x - x₀) + b(y - y₀) + c(z - z₀) = 0 |
Students often think the dot product gives a vector. It does not. The dot product always returns a scalar (a single number). The cross product is the operation that returns a vector.
Students often swap the order of a cross product and expect the same result. a × b = -(b × a). Reversing the order flips the sign. The dot product, by contrast, is commutative.
Students sometimes confuse "parallel" and "orthogonal" conditions. Parallel vectors satisfy a · b = ±|a||b| (cos θ = ±1). Orthogonal vectors satisfy a · b = 0 (cos θ = 0). For the cross product it is the opposite: parallel vectors give a × b = 0, while perpendicular vectors give the maximum magnitude.
Students often forget that two lines in 3D can be skew (neither parallel nor intersecting). In 2D this cannot happen, which is why it catches people off guard.
⚠️ Computing the cross product via the determinant is tested heavily. Practise the 3×3 expansion until it is mechanical.
⚠️ "Find the equation of a plane given three points" is a classic exam question. The method: form two vectors, cross them for the normal, then plug in.
⚠️ Identifying quadric surfaces from their equations comes up frequently. Know the sign patterns: two positives and one negative on the left with = 1 is a hyperboloid of one sheet; one positive and two negatives with = -1 is two sheets.
⚠️ Projection problems (find the component of b in the direction of a, or the vector projection) appear regularly. Memorise both formulas and know which returns a number versus a vector.
⚠️ Distance formula and sphere equations are the simplest points on any exam. Do not lose them to algebraic slips.
True or false: the dot product of two vectors can be negative.
True. When the angle between them is greater than 90°, cos θ is negative.
Fill in the blank: if a × b = ⟨0, 0, 0⟩ and neither vector is zero, then a and b are ______.
Parallel.
True or false: two lines in 3D that do not intersect must be parallel.
False. They could be skew.
Fill in the blank: the xy-plane is the set of all points where ______ = 0.
z.
True or false: |a × b| gives the area of the triangle formed by a and b.
False. It gives the area of the parallelogram. The triangle's area is half that.
Q: Find the equation of the plane through the point (1, 2, 3) with normal vector ⟨4, -1, 2⟩.
A: 4(x - 1) - 1(y - 2) + 2(z - 3) = 0, which simplifies to 4x - y + 2z = 8.
Q: Compute a × b where a = ⟨1, 0, 3⟩ and b = ⟨2, 1, -1⟩.
A: a × b = ⟨(0)(-1) - (3)(1), (3)(2) - (1)(-1), (1)(1) - (0)(2)⟩ = ⟨-3, 7, 1⟩.
Q: Find the vector projection of b = ⟨3, 4⟩ onto a = ⟨1, 0⟩.
A: a · b = 3, |a|² = 1, so proj_a(b) = (3/1)⟨1, 0⟩ = ⟨3, 0⟩.
Q: Are the vectors ⟨2, -1, 5⟩ and ⟨1, 3, 1⟩ orthogonal?
A: Dot product = 2(1) + (-1)(3) + 5(1) = 2 - 3 + 5 = 4. Since 4 ≠ 0, they are not orthogonal.
Q: Classify the surface x² + y² - z² = 1.
A: Two positive squared terms and one negative squared term, equal to 1. This is a hyperboloid of one sheet.
The dot product and cross product from this chapter are the building blocks for everything in Chapters 13 and 14. Tangent vectors, normal vectors, curvature, and tangent planes all rely on these operations.
Lines and planes reappear when you set up tangent planes to surfaces (Chapter 14) and when you describe the intersection curves of two surfaces (Chapter 13). The quadric surfaces return as the domains and boundaries for double integrals in Chapter 15.
Related Terms / Search Tags
vectors in 3D, dot product, scalar product, cross product, vector product, magnitude, unit vector, orthogonal vectors, perpendicular vectors, projection, component projection, vector projection, parametric equations, symmetric equations, direction vector, normal vector, plane equation, quadric surfaces, ellipsoid, hyperboloid, paraboloid, cone, cylinder, traces, MATH 126, Calculus III, UW, multivariable calculus, 3D geometry