Source: Midterm I Practice Exam
Difficulty: Intermediate | Prerequisites: Vectors, dot products, and cross products (see companion notes). Familiarity with standard conic sections (ellipses, parabolas) helps.
Planes and lines in 3D space are the geometric building blocks you use throughout multivariable calculus. Knowing how to find where planes intersect, how to write a plane through given points and lines, and how to recognise and write equations for common surfaces (ellipsoids, cylinders, spheres) is tested directly on Midterm I. This material bridges the vector algebra from the first few lectures and the parametric curves that follow. If you can handle these problems, you have the spatial reasoning the rest of the course depends on.
Two non-parallel planes intersect in a line; find it by cross-producting the normal vectors (for direction) and solving the system (for a point). Quadric surfaces are the 3D analogues of conic sections: ellipsoids, cylinders, and so on. Tangent spheres touch at exactly one point, and you find the radius by setting the distance between centres equal to the sum or difference of the radii.
Normal vector of a plane
A vector perpendicular to every vector lying in the plane. For Ax + By + Cz = D, the normal vector is n = ⟨A, B, C⟩.
In simple terms, this means: the arrow that sticks straight out of the plane.
Line of intersection of two planes
The set of points that lie on both planes simultaneously. It is a straight line whose direction is perpendicular to both normal vectors.
Think of it as: the crease where two sheets of paper meet.
Ellipsoid
A surface of the form x²/a² + y²/b² + z²/c² = 1. It looks like a stretched or compressed sphere.
In simple terms, this means: a 3D oval, like an egg.
Parabolic cylinder
A surface defined by a parabolic equation in two variables (e.g. z = x²), with the third variable free. It extends infinitely in the direction of the missing variable.
Think of it as: a parabola extruded into a trough shape.
Elliptic cylinder
A surface of the form x²/a² + z²/c² = 1 (or any pair of variables). The missing variable is free, so the ellipse extends as a tube.
Helix
A curve that spirals around a cylinder. Parametrically: x = a cos t, y = t, z = b sin t (wrapping around a circular cylinder), with variations for elliptic cylinders.
Tangent spheres
Two spheres that touch at exactly one point. They are either externally tangent (outsides touch) or internally tangent (one sits inside the other, touching at one point).
Given planes 2x - 3y = 9 and x + 2y - z = -4:
Step 1: Find the direction vector. The direction of the line is perpendicular to both normal vectors. Normal to plane 1: n₁ = ⟨2, -3, 0⟩. Normal to plane 2: n₂ = ⟨1, 2, -1⟩.
Direction = n₁ × n₂ = ⟨(-3)(-1) - (0)(2), (0)(1) - (2)(-1), (2)(2) - (-3)(1)⟩ = ⟨3, 2, 7⟩.
Step 2: Find a point on the line. Set one variable to a convenient value (e.g. z = 0, or y = 0). With y = 0: from 2x = 9, x = 9/2. From x + 0 - z = -4, z = 9/2 + 4 = 17/2. Point: (9/2, 0, 17/2).
Step 3: Write the vector equation. r(t) = ⟨9/2, 0, 17/2⟩ + t⟨3, 2, 7⟩.
Verification: Substitute back into both plane equations. At t = 0: 2(9/2) - 3(0) = 9. ✓ And (9/2) + 2(0) - (17/2) = 9/2 - 17/2 = -8/2 = -4. ✓
Given the line above and the point (3, 1, -4), find the plane Ax + By + Cz = D.
You need two vectors in the plane: the direction of the line ⟨3, 2, 7⟩ and a vector from the known point to a point on the line, e.g. (9/2, 0, 17/2) - (3, 1, -4) = ⟨3/2, -1, 25/2⟩.
The normal to the new plane is the cross product of these two vectors.
n = ⟨3, 2, 7⟩ × ⟨3/2, -1, 25/2⟩ = ⟨(2)(25/2) - (7)(-1), (7)(3/2) - (3)(25/2), (3)(-1) - (2)(3/2)⟩ = ⟨25 + 7, 21/2 - 75/2, -3 - 3⟩ = ⟨32, -27, -6⟩.
Using point (3, 1, -4): 32(3) + (-27)(1) + (-6)(-4) = 96 - 27 + 24 = 93.
Plane equation: 32x - 27y - 6z = 93.
An ostrich egg with circumference 34 cm and height 20 cm, centred at the origin with z-axis vertical.
Height = 20 cm, so the semi-axis along z is c = 10.
Circumference = 34 cm = 2πr, so r = 34/(2π) = 17/π. This gives a = b = 17/π.
Equation: x²/(17/π)² + y²/(17/π)² + z²/10² = 1, or π²x²/289 + π²y²/289 + z²/100 = 1.
Symmetric about the yz-plane means the equation is unchanged when x is replaced by -x. So x appears only as x².
Any equation of the form y = x² or z = x² (or y = -x², etc.) works. The variable not in the equation is free, creating the "cylinder" extension.
Example: y = x². This is a parabola in the xy-plane, extended infinitely along z.
The cylinder is x²/4 + z²/9 = 1. The cross-section is an ellipse with semi-axes a = 2 (in x) and c = 3 (in z). The free variable is y.
Parametrise the ellipse: x = 2 cos t, z = 3 sin t.
Let y increase linearly: y = t.
Parametric equations: x(t) = 2 cos t, y(t) = t, z(t) = 3 sin t.
Sphere 1: (x - 5)² + (y + 3)² + (z - 6)² = 4. Centre C₁ = (5, -3, 6), radius r₁ = 2.
Sphere 2: x² + y² + z² = r². Centre C₂ = (0, 0, 0), radius r₂ = r.
Distance between centres: d = √(25 + 9 + 36) = √70.
For tangent spheres, the distance between centres equals either the sum or the difference of the radii:
External tangency: d = r₁ + r₂, so r = √70 - 2.
Internal tangency: d = |r₂ - r₁|, so r = √70 + 2.
\text{Plane: } Ax + By + Cz = D, \quad \mathbf{n} = \langle A, B, C \rangle\text{Direction of line of intersection} = \mathbf{n}_1 \times \mathbf{n}_2\text{Ellipsoid: } \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\text{Distance between centres: } d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\text{External tangency: } d = r_1 + r_2 \qquad \text{Internal tangency: } d = |r_1 - r_2|Ellipsoids model everything from eggs to planetary shapes (Earth is an oblate ellipsoid). Parabolic cylinders appear in the design of satellite dishes and solar troughs. The intersection of planes is how CAD software computes edges of 3D objects, and tangent-sphere geometry is used in molecular modelling and collision detection in physics engines.
Students often forget that the first plane, 2x - 3y = 9, has an implicit z coefficient of 0. Its normal vector is ⟨2, -3, 0⟩, not ⟨2, -3⟩.
When finding a point on the line of intersection, students sometimes set two variables and over-constrain the system. Set only one free variable and solve the remaining system of two equations in two unknowns.
For the tangent spheres problem, students frequently find only one value of r (usually the external tangency case) and forget the internal tangency case. There are always two answers.
A "cylinder" in multivariable calculus is not necessarily circular. Any curve extended along a free axis is a cylinder. A parabolic cylinder is a parabola extended into 3D.
⚠️ Always verify your line of intersection lies on both planes. Substitute back. This is explicitly requested on the exam and catches algebra errors.
⚠️ For the ellipsoid problem, the circumference gives you the radius of the widest circle, not the diameter. Convert carefully: C = 2πr.
⚠️ The parabolic cylinder must be symmetric about the yz-plane. That means x can only appear as x² (even power). Writing y = x or z = x would fail the symmetry condition.
⚠️ Tangent spheres: there are always two values of r. Missing one costs half the marks.
The normal vector to the plane 3x - y + 5z = 12 is ___. (⟨3, -1, 5⟩.)
True or false: the direction of the line of intersection of two planes is found by dotting the two normals. (False. You cross them.)
An ellipsoid centred at the origin with semi-axes 3, 4, 5 has equation ___. (x²/9 + y²/16 + z²/25 = 1.)
True or false: a parabolic cylinder in 3D extends infinitely in one direction. (True.)
Two spheres with radii 3 and 5 whose centres are distance 8 apart are ___. (Externally tangent.)
Q: Find the line of intersection of the planes x + y + z = 6 and 2x - y = 3.
A: Normals: ⟨1, 1, 1⟩ and ⟨2, -1, 0⟩. Direction = ⟨1, 1, 1⟩ × ⟨2, -1, 0⟩ = ⟨(1)(0) - (1)(-1), (1)(2) - (1)(0), (1)(-1) - (1)(2)⟩ = ⟨1, 2, -3⟩. Set z = 0: x + y = 6, 2x - y = 3. Adding: 3x = 9, x = 3, y = 3. Point (3, 3, 0). Line: r(t) = ⟨3, 3, 0⟩ + t⟨1, 2, -3⟩.
Q: Write the equation of an ellipsoid centred at the origin with semi-axes 2 along x, 3 along y, and 5 along z.
A: x²/4 + y²/9 + z²/25 = 1.
Q: Two spheres have centres (1, 2, 3) and (4, 6, 3) with radii 2 and r. If they are tangent, find r.
A: Distance = √(9 + 16 + 0) = 5. External: r = 5 - 2 = 3. Internal: r = 5 + 2 = 7.
Q: Write parametric equations for a helix around the circular cylinder x² + z² = 9.
A: x(t) = 3 cos t, z(t) = 3 sin t, y(t) = t (or any linear function of t).
Normal vectors to planes connect directly to the gradient vector in later chapters. The parametric helix reappears when you study arc length and curvature of space curves (see the third set of these study notes). Tangent-sphere problems build the same distance-and-radius reasoning used in collision detection and optimisation problems.
Tags: planes in 3D, line of intersection, normal vector, cross product of normals, ellipsoid equation, parabolic cylinder, elliptic cylinder, helix parametric equations, tangent spheres, internally tangent, externally tangent, quadric surfaces, Math 126, Calculus III, UW, multivariable calculus