Difficulty: Intermediate | Prerequisites: Calculus II, basic vector operations (dot product, cross product)
This unit covers how to describe lines, planes, and surfaces in three-dimensional space using vectors. You will classify pairs of lines (parallel, intersecting, or skew), find where a line meets a plane, identify quadric surfaces from their equations, and use the cross product and dot product to compute areas, volumes, and distances. These tools underpin nearly everything in the rest of Calculus III.
Direction vector
A vector that indicates the direction a line travels. Two lines are parallel when their direction vectors are scalar multiples of each other. Think of it as the arrow you follow if you walk along the line.
Parametric equations of a line
Expressions of the form x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where (x₀, y₀, z₀) is a point on the line and ⟨a, b, c⟩ is the direction vector. In simple terms, plugging in different values of t walks you along the line.
Skew lines
Lines that are neither parallel nor intersecting; they exist in different planes in 3D. Think of it as two roads on different levels of a motorway interchange, never touching and never running the same way.
Quadric surface
A surface in 3D described by a second-degree equation in x, y, z. These are the 3D analogues of conic sections (ellipses, parabolas, hyperbolas).
Ellipsoid
A surface of the form x²/a² + y²/b² + z²/c² = 1. All cross-sections are ellipses. When a = b = c, it is a sphere.
Hyperboloid of one sheet
A surface of the form x²/a² + y²/b² − z²/c² = 1. It looks like a cooling tower: one connected piece with a waist in the middle. Horizontal cross-sections are ellipses; vertical cross-sections are hyperbolas.
Hyperboloid of two sheets
A surface of the form −x²/a² − y²/b² + z²/c² = 1. Two separate bowl-shaped pieces facing away from each other.
Elliptic paraboloid
A surface of the form z = x²/a² + y²/b² (or z/c = ...). It looks like an upward-opening bowl. Horizontal cross-sections are ellipses; vertical cross-sections are parabolas.
Hyperbolic paraboloid
A surface of the form z = x²/a² − y²/b². A saddle shape: it curves up in one direction and down in the other.
Cone
A surface of the form z²/c² = x²/a² + y²/b². The tip is at the origin. Horizontal cross-sections are ellipses; vertical cross-sections are hyperbolas.
Scalar triple product
The value a · (b × c), which equals the volume of the parallelepiped formed by vectors a, b, c. In simple terms, it measures the 3D "box" that three vectors span.
Point-to-plane distance
The perpendicular distance from a point (x₁, y₁, z₁) to the plane Ax + By + Cz = D, given by |Ax₁ + By₁ + Cz₁ − D| / √(A² + B² + C²).
Given two lines in parametric form, determine whether they are parallel, intersecting, or skew by following these steps:
Check direction vectors. If one direction vector is a scalar multiple of the other, the lines are parallel (they may also be coincident).
If not parallel, solve for an intersection. Set the parametric equations equal (using different parameters for each line, say t and s) and solve the resulting system. If a consistent solution exists, the lines intersect at that point.
If no consistent solution, the lines are skew. They pass each other without touching.
Worked example from the cheat sheet (Section 12.5):
L₁: ⟨2 − 3t, 4 − t, −6 + 4t⟩ and L₂: ⟨4 + 8t, 1 − t, 4 + 2t⟩
Direction vectors: ⟨−3, −1, 4⟩ and ⟨8, −1, 2⟩. Not scalar multiples, so not parallel.
Setting components equal (using parameter s for L₂): solve the system. The cheat sheet finds t = 2, confirming intersection.
When two planes intersect, their line of intersection can be found by:
Cross the normal vectors of the two planes to get the direction vector of the line.
Find a specific point on both planes by setting one variable (e.g. x = t) and solving the remaining system.
Example from the cheat sheet:
Planes 4x − y + z = 2 and x − 3y = 4. The cheat sheet parametrises with x = t, then solves for y and z in terms of t, yielding the line z = 2/3 − 11/3 t (with x = t).
Quadric surfaces are identified by their general equation and their cross-sections (traces). The strategy is always the same: set one variable equal to a constant and see what curve you get.
x²/a² + y²/b² + z²/c² = 1
All traces are ellipses (or circles when two denominators match).
When a = b = c, you have a sphere.
z²/c² = x²/a² + y²/b²
Horizontal traces (z = k, k ≠ 0): ellipses.
Vertical traces (x = 0 or y = 0): pairs of lines through the origin (two crossing lines, forming the classic "X" shape).
The vertex is at the origin.
x²/a² + y²/b² − z²/c² = 1
Horizontal traces: ellipses.
Vertical traces: hyperbolas.
One connected surface. The cheat sheet notes: when k > 0 or < 0 in the constant, horizontal traces are ellipses and vertical traces are hyperbolas.
−x²/a² − y²/b² + z²/c² = 1
Horizontal traces: ellipses (only for |z| > c).
Vertical traces: hyperbolas.
Two separate pieces.
z/c = x²/a² + y²/b²
Horizontal traces: ellipses.
Vertical traces: parabolas.
z/c = x²/a² − y²/b²
Horizontal traces: hyperbolas.
Vertical traces: parabolas.
The saddle surface.
For any equation, set each variable to a constant k in turn and identify the resulting 2D curve. This tells you the surface type. The cheat sheet labels these as "Horizontal" and "Vertical" traces for each surface.
The area of the parallelogram spanned by vectors a and b is |a × b|. If you need the area of a triangle, take half.
The volume of the box spanned by vectors a, b, c is |a · (b × c)| (the absolute value of the scalar triple product).
Given the plane Ax + By + Cz = D and a point (x₁, y₁, z₁):
d = |Ax₁ + By₁ + Cz₁ − D| / √(A² + B² + C²)
The normal vector to the plane is ⟨A, B, C⟩. This formula appears prominently on the cheat sheet and is a standard exam question.
The cheat sheet also notes that distance = comp_a b, the scalar projection of one vector onto another. This is equivalent: projecting the vector from any point on the plane to the external point onto the plane's normal gives the perpendicular distance.
Area of parallelogram: |a × b|
Volume of parallelepiped: V = |a · (b × c)|
Point-to-plane distance: d = |Ax₁ + By₁ + Cz₁ − D| / √(A² + B² + C²)
Parametric line: x = x₀ + at, y = y₀ + bt, z = z₀ + ct
Ellipsoid: x²/a² + y²/b² + z²/c² = 1
Cone: z²/c² = x²/a² + y²/b²
Hyperboloid of one sheet: x²/a² + y²/b² − z²/c² = 1
Hyperboloid of two sheets: −x²/a² − y²/b² + z²/c² = 1
Elliptic paraboloid: z/c = x²/a² + y²/b²
Hyperbolic paraboloid: z/c = x²/a² − y²/b²
Students often assume that two non-parallel lines in 3D must intersect. They do not. In three dimensions, lines can be skew: neither parallel nor intersecting.
Confusing the hyperboloid of one sheet with two sheets. The one-sheet version has the positive terms outnumbering the negative (two positive, one negative). The two-sheet version has two negative terms and one positive.
Forgetting to take the absolute value when computing volumes or distances. The scalar triple product and the distance formula both require absolute value.
Mixing up the cross product (which gives a vector) with the dot product (which gives a scalar). Area uses the magnitude of the cross product; projection uses the dot product.
⚠️ Classifying lines (parallel vs intersecting vs skew) is a bread-and-butter exam problem. You will almost certainly see one.
⚠️ Identifying a quadric surface from its equation and sketching its traces is frequently tested. Know the sign patterns.
⚠️ The point-to-plane distance formula appears regularly. Memorise it and be comfortable plugging in.
⚠️ Cross-product applications (area and volume) are quick marks if you know the formulas, easy marks lost if you confuse cross and dot products.
True or false: Two lines in 3D that are not parallel must intersect. (False, they can be skew.)
Fill in the blank: The surface x²/4 + y²/9 + z²/16 = 1 is called an ________. (Ellipsoid.)
True or false: The cross product of two vectors is a scalar. (False, it is a vector.)
Fill in the blank: The volume of a parallelepiped formed by vectors a, b, c is |a · (______)|. (b × c)
True or false: A hyperbolic paraboloid has elliptical horizontal cross-sections. (False, they are hyperbolas.)
Q: Given L₁ with direction vector ⟨2, −1, 3⟩ and L₂ with direction vector ⟨−4, 2, −6⟩, are the lines parallel, intersecting, or skew?
A: Parallel. ⟨−4, 2, −6⟩ = −2 × ⟨2, −1, 3⟩, so the direction vectors are scalar multiples.
Q: What type of surface is described by x² + y²/4 − z²/9 = 1?
A: Hyperboloid of one sheet. Two positive terms and one negative term with the equation equalling 1.
Q: Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 4.
A: d = |2(1) − 1(2) + 2(3) − 4| / √(4 + 1 + 4) = |2 − 2 + 6 − 4| / 3 = 2/3.
Q: Vectors a = ⟨1, 0, 0⟩ and b = ⟨0, 1, 0⟩ span a parallelogram. What is its area?
A: a × b = ⟨0, 0, 1⟩. |a × b| = 1. The area is 1.
Q: How do you determine whether the surface z² = x² + y² is a cone or a hyperboloid?
A: It is a cone. The equation can be written z² − x² − y² = 0 (equals zero, not 1), and the vertex is at the origin. A hyperboloid equals 1 or −1.
The cross product and dot product reappear throughout Calculus III: curvature formulas rely on |r'(t) × r''(t)|, and line integrals use dot products of vector fields with tangent vectors. Quadric surfaces are the surfaces you will parametrise later when computing surface integrals. The point-to-plane distance formula is a special case of projection, which generalises to distances between skew lines and between parallel planes.
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