Difficulty: Intermediate | Prerequisites: Vectors, dot products, cross products (MATH 125 or equivalent). You should be comfortable with basic vector arithmetic and equations of lines before starting here.
This material covers the geometric backbone of MATH 126 (Calculus III): working with lines and planes in three dimensions, and identifying quadric surfaces from their equations. These topics appear early in the course and underpin everything that follows, from partial derivatives to surface integrals. If you missed the first few weeks, this is the foundation you need. You should already be comfortable with vectors, dot products, and cross products from MATH 125.
Lines in 3D are described by parametric equations using a point and a direction vector. Planes are defined by a normal vector and a point, and you can find where a line meets a plane by substitution. Quadric surfaces are the 3D analogues of conic sections, and you identify them by completing the square and comparing to standard forms.
Parametric equations of a line
A way of expressing a line in 3D using a known point (x₀, y₀, z₀) and a direction vector ⟨a, b, c⟩: x = x₀ + at, y = y₀ + bt, z = z₀ + ct. Think of it as a recipe for generating every point on the line by sliding the parameter t.
Normal vector
A vector perpendicular to a plane. For the plane Ax + By + Cz = D, the normal vector is ⟨A, B, C⟩. In simple terms, it points straight "out" from the flat surface.
Direction vector
A vector that runs along a line, telling you which way the line points. Any scalar multiple of a direction vector works equally well.
Quadric surface
A surface in 3D defined by a second-degree equation in x, y, and z. These are the 3D versions of conic sections (ellipses, parabolas, hyperbolas). Common types include ellipsoids, paraboloids, hyperboloids, and cones.
Circular (elliptical) paraboloid
A bowl-shaped quadric surface. In standard form it looks like z = x² + y² (circular) or z = x²/a² + y²/b² (elliptical). Think of it as the shape you get when you spin a parabola around its axis.
A line needs two ingredients: a point it passes through and a direction vector.
Given point P₀ = (x₀, y₀, z₀) and direction ⟨a, b, c⟩, the parametric equations are:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
To find a direction vector between two points W and R, compute the vector RW = W − R (or R − W; either direction works, you just get opposite signs).
The normal vector of the plane serves as the direction vector of the line.
Example: For the plane −x + y + z = 2, the normal vector is ⟨−1, 1, 1⟩.
A line through point Q orthogonal to the plane uses ⟨−1, 1, 1⟩ as its direction.
Substitute the parametric equations of the line into the plane equation.
Solve for t, then plug t back in to get the coordinates of the intersection point.
Example: Line L₁ through (2, 1, −1) with direction ⟨−1, 1, 1⟩. Substituting into −x + y + z = 2 gives −(2 − t) + (1 + t) + (−1 + t) = 2, which simplifies to −2 + 3t = 2, so t = 4/3.
Intersection point R = (2/3, 7/3, 1/3).
Compute the direction vector from the two points.
Use either point as the base point in the parametric equations.
Example: Points W = (−1, 3, 1) and R = (2/3, 7/3, 1/3). Direction RW = ⟨−5/3, 2/3, 2/3⟩. You can scale this to ⟨−5, 2, 2⟩ for cleaner numbers.
Parametric equations: x = −1 − 5t, y = 3 + 2t, z = 1 + 2t.
Start by completing the square where needed and rearranging into a standard form.
The key question: which variables appear squared, and what are the signs?
"All points equidistant from a plane and a point" is the definition of a paraboloid (analogous to the 2D definition of a parabola as equidistant from a line and a focus).
Set the distance to the plane equal to the distance to the point and simplify.
Example: Equidistant from y = 3 and (0, 1, 2). Distance to the plane: |y − 3|. Distance to the point: √(x² + (y − 1)² + (z − 2)²). Squaring both sides and expanding gives −4(y − 2) = x² + (z − 2)², which is a circular (elliptical) paraboloid opening in the −y direction.
Ellipsoid: x²/a² + y²/b² + z²/c² = 1 (all positive, all squared)
Elliptic paraboloid: z = x²/a² + y²/b² (one variable linear, others squared, same sign)
Hyperbolic paraboloid: z = x²/a² − y²/b² (saddle shape, mixed signs)
Cone: x²/a² + y²/b² = z²/c² (looks like a paraboloid but all three variables squared)
Hyperboloid of one sheet: x²/a² + y²/b² − z²/c² = 1 (one negative sign)
Hyperboloid of two sheets: −x²/a² − y²/b² + z²/c² = 1 (two negative signs)
Parametric line through P₀ with direction ⟨a, b, c⟩
x = x₀ + at, y = y₀ + bt, z = z₀ + ct
Equation of a plane with normal ⟨A, B, C⟩ through point (x₀, y₀, z₀)
A(x − x₀) + B(y − y₀) + C(z − z₀) = 0
Distance from point (x₁, y₁, z₁) to point (x₂, y₂, z₂)
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
Distance from point (x₀, y₀, z₀) to plane Ax + By + Cz = D
d = |Ax₀ + By₀ + Cz₀ − D| / √(A² + B² + C²)
Parametric lines are used in computer graphics for ray tracing: every ray of light from a virtual camera is a parametric line, and finding where it hits a surface is exactly the line-plane intersection problem. Quadric surfaces show up in satellite dish design (paraboloids focus signals to a single point) and in architectural structures like cooling towers (hyperboloids of one sheet).
Students often confuse the normal vector of a plane with a direction vector of a line in that plane. The normal is perpendicular to the plane, not along it.
When finding a line orthogonal to a plane, some students try to use a cross product. You do not need one here: the normal vector of the plane is already the direction you want.
Students sometimes forget to square both sides when setting up the equidistant condition for quadric surfaces. If you leave the square root on one side, the algebra will not simplify properly.
A cone and a paraboloid can look similar in cross-section, but a cone has all three variables squared, while a paraboloid has one variable appearing linearly.
⚠️ Line-plane intersection problems appear on nearly every MATH 126 final. The procedure is always the same: substitute, solve for t, plug back in.
⚠️ You will be asked to identify quadric surfaces by name. Practise completing the square and matching to standard forms until it is automatic.
⚠️ Direction vectors can be scaled freely. If your answer has fractions, multiply through by the common denominator for cleaner parametric equations.
⚠️ "Find parametric equations" means give x, y, z each as a function of t. Forgetting to state all three is a common mark-losing error.
True or False: The normal vector of the plane 3x − 2y + z = 7 is ⟨3, −2, 1⟩. (True)
Fill in the blank: To find where a line meets a plane, substitute the ______ equations into the plane equation and solve for t. (parametric)
True or False: A paraboloid has all three variables appearing as squared terms. (False, one variable is linear)
Fill in the blank: A line perpendicular to a plane has its direction vector equal to the plane's ______ vector. (normal)
True or False: Scaling a direction vector by a nonzero constant changes the line it describes. (False, same line, different parametrisation)
Q: The plane P has equation −x + y + z = 2. Point Q = (2, 1, −1) is not on P. Write parametric equations for the line through Q perpendicular to P.
A: The normal vector of P is ⟨−1, 1, 1⟩, which serves as the direction. Parametric equations: x = 2 − t, y = 1 + t, z = −1 + t.
Q: Find the point where the line x = 2 − t, y = 1 + t, z = −1 + t intersects the plane −x + y + z = 2.
A: Substitute: −(2 − t) + (1 + t) + (−1 + t) = 2 → −2 + 3t = 2 → t = 4/3. Point R = (2/3, 7/3, 1/3).
Q: Find the equation of the quadric surface consisting of all points equidistant from the plane y = 3 and the point (0, 1, 2).
A: Set |y − 3| = √(x² + (y − 1)² + (z − 2)²). Square both sides: (y − 3)² = x² + (y − 1)² + (z − 2)². Expand and simplify to get −4(y − 2) = x² + (z − 2)². This is a circular paraboloid.
Q: Identify the quadric surface given by the equation 4x² + y² − z² = 4.
A: Divide through by 4: x² + y²/4 − z²/4 = 1. Two positive squared terms and one negative: this is a hyperboloid of one sheet.
This connects to partial derivatives and tangent planes because the normal vector of a surface at a point comes from the gradient, which is built from partial derivatives. It also connects to curvature, since understanding curves in 3D requires parametric representations. The quadric surface identification skills return when you study Lagrange multipliers (you need to recognise constraint surfaces) and again in surface integrals.
Parametric equations, vector equation of a line, direction vector, normal vector, equation of a plane, line-plane intersection, quadric surfaces, ellipsoid, paraboloid, circular paraboloid, elliptic paraboloid, hyperbolic paraboloid, saddle surface, hyperboloid of one sheet, hyperboloid of two sheets, cone, completing the square in 3D, equidistant surface, conic sections in 3D, MATH 126, Calculus III, UW