Difficulty: Intermediate | Prerequisites: Vector operations, partial derivatives
Planes in 3D are defined by linear equations, and their normal vectors come straight from the coefficients. The tangent plane to a surface z = f(x, y) at a point is the best linear approximation of the surface there, built from partial derivatives. Quadratic forms (expressions like ax² + bxy + cy²) determine whether a surface looks like a bowl, an inverted bowl, or a saddle.
Level set
The set of all points (x, y, z) where a function takes a single constant value. For a linear function like T(x, y, z) = 2x + y + 3z + 27, each level set is a plane in space. Think of it as a contour line, but in three dimensions.
Normal vector
A vector perpendicular to a surface or plane at a given point. For the plane Ax + By + Cz + D = 0, the normal vector is n = ⟨A, B, C⟩. In simple terms, it points straight out from the surface.
Tangent plane
The plane that just touches a surface z = f(x, y) at a point and best approximates the surface near that point. It is the 3D analogue of a tangent line to a curve.
Partial derivative
The derivative of a multivariable function with respect to one variable, treating all others as constants. f_x is the rate of change of f in the x-direction; f_y is the rate of change in the y-direction.
Quadratic form
An expression of the type ax² + bxy + cy², where a, b, c are constants. The sign pattern of the eigenvalues (or, equivalently, the discriminant ac − (b/2)²) determines the surface shape.
Paraboloid
A surface shaped like a bowl (opening upward) or an inverted bowl (opening downward). It arises when the quadratic form is positive definite or negative definite.
Saddle (hyperbolic paraboloid)
A surface that curves upward in one direction and downward in another, like the middle of a horse saddle. It arises when the quadratic form is indefinite (mixed signs).
A linear function T(x, y, z) = Ax + By + Cz + D has level sets of the form Ax + By + Cz + D = c, where c is a constant.
Each level set is a plane. Changing c shifts the plane parallel to itself.
The normal vector to every level set is n = ⟨A, B, C⟩, read directly from the coefficients.
Example from the course: T(x, y, z) = 2x + y + 3z + 27 has level sets 2x + y + 3z + 27 = T₀, all parallel planes with normal ⟨2, 1, 3⟩.
For a plane Ax + By + Cz + D = 0 and a point (x₀, y₀, z₀):
d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
This formula measures the perpendicular (shortest) distance.
A typical exam application: finding how far a point (say a bug at (0, 1, 2)) is from a temperature level set.
For a surface z = f(x, y), the tangent plane at the point (x₀, y₀, f(x₀, y₀)) is:
z = f(x₀, y₀) + f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀)
Equivalently: z − f(x₀, y₀) = f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀).
The tangent plane is the linear approximation of the surface near that point. It captures the first-order behaviour.
The normal vector to the tangent plane is ⟨f_x, f_y, −1⟩ (or ⟨−f_x, −f_y, 1⟩, depending on orientation).
When a surface z = f(x, y) is intersected by a vertical plane (e.g. x = 2), you get a curve on the surface.
To find the tangent line to that curve at a point: restrict f to that plane (substitute x = 2), differentiate with respect to the remaining variable, and evaluate.
The tangent line lies in the cutting plane and touches the curve at the specified point.
f(x, y) = x² + 6xy + y²: the cross-term 6xy is large relative to the squared terms. Check the discriminant: ac − (b/2)² = (1)(1) − 9 = −8 < 0. This is a saddle, despite appearances.
g(x, y) = −4x² − y² + 4xy: negative leading terms with a cross-term. Discriminant: (−4)(−1) − (4/2)² = 4 − 4 = 0. This is degenerate (borderline), but the negative leading coefficients suggest an inverted or saddle shape.
h(x, y) = 7x² + 2xy + 4y²: discriminant: (7)(4) − (2/2)² = 28 − 1 = 27 > 0, and a > 0. This is positive definite: an upward-opening paraboloid.
Rule of thumb: if the discriminant ac − (b/2)² is positive and a > 0, the surface opens upward. If positive and a < 0, it opens downward. If negative, it is a saddle.
d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\text{Normal to tangent plane:} \quad \mathbf{n} = \langle f_x,\; f_y,\; -1 \rangle\text{Discriminant of } ax^2 + bxy + cy^2: \quad ac - \left(\frac{b}{2}\right)^2Students often forget to put the plane equation in the form Ax + By + Cz + D = 0 before applying the point-to-plane distance formula. If your equation is 2x + y + 3z + 27 = 30, rewrite it as 2x + y + 3z − 3 = 0 first.
A common error in tangent plane problems: using f(x₀, y₀) where you need f_x(x₀, y₀). The tangent plane formula requires partial derivatives, not the function value alone (though the function value does appear as the constant term).
Students sometimes identify a quadratic form as a paraboloid just because it has x² and y² terms. The cross-term bxy matters: if it is large enough, the surface is a saddle. Always check the discriminant.
The normal vector to the tangent plane is ⟨f_x, f_y, −1⟩, not ⟨f_x, f_y, 0⟩. The −1 comes from the z-component of the surface equation z − f(x, y) = 0.
⚠️ The tangent plane formula is one of the most commonly tested items on Midterm 1. Be able to compute f_x and f_y, evaluate them at a point, and write down the tangent plane equation in one go.
⚠️ Point-to-plane distance appears regularly. Make sure you can identify A, B, C, D from the equation and plug into the formula without sign errors.
⚠️ Matching quadratic forms to surface graphs is a staple of the exam. The discriminant test is quick and reliable.
⚠️ Tangent line to a curve on a surface (the intersection of a surface with a plane like x = 2) is a common follow-up to a tangent plane question. Restrict, differentiate, evaluate.
Fill in the blank: the normal vector to the plane 3x − 2y + 5z = 7 is ⟨____⟩.
⟨3, −2, 5⟩.
True or false: the tangent plane at a point uses the second derivatives of f.
False. It uses only first partial derivatives (f_x and f_y).
Fill in the blank: a quadratic form with discriminant ac − (b/2)² > 0 and a > 0 gives an upward-opening ______.
Paraboloid.
True or false: if the discriminant of a quadratic form is negative, the surface is a saddle.
True.
Fill in the blank: in the point-to-plane distance formula, the denominator is √(________).
A² + B² + C².
Q: Find the distance from the point (0, 1, 2) to the plane 2x + y + 3z − 3 = 0.
A: d = |2(0) + 1(1) + 3(2) − 3| / √(4 + 1 + 9) = |0 + 1 + 6 − 3| / √14 = 4/√14 = 4√14/14 = 2√14/7.
Q: Find the tangent plane to z = x² + y² at the point (1, 2, 5).
A: f_x = 2x, so f_x(1, 2) = 2. f_y = 2y, so f_y(1, 2) = 4. Tangent plane: z = 5 + 2(x − 1) + 4(y − 2) = 2x + 4y − 5.
Q: For f(x, y) = x² + 6xy + y², is the surface at z = f(x, y) a paraboloid or a saddle?
A: a = 1, b = 6, c = 1. Discriminant: ac − (b/2)² = 1 − 9 = −8 < 0. The surface is a saddle (hyperbolic paraboloid).
Q: Find the equation of the tangent plane to z = 3x²y at the point (1, 2, 6).
A: f_x = 6xy, so f_x(1, 2) = 12. f_y = 3x², so f_y(1, 2) = 3. Tangent plane: z = 6 + 12(x − 1) + 3(y − 2) = 12x + 3y − 12.
Q: The surface z = f(x, y) is cut by the plane x = 2. If f(x, y) = x² + y³, find the slope of the tangent line to the resulting curve at y = 1.
A: Restricting to x = 2: z = 4 + y³. dz/dy = 3y². At y = 1: slope = 3. The tangent line at (2, 1, 5) has direction ⟨0, 1, 3⟩.
The normal vector to a plane connects directly to the dot product and cross product from the Vector Operations notes: if you know two direction vectors in a plane, their cross product gives the normal. The tangent plane reappears when you study the gradient later in the course, since the gradient of f at a point is perpendicular to the level curve through that point.
Quadratic forms and the discriminant test are a preview of the second derivative test for critical points, which classifies local maxima, minima, and saddle points using the Hessian matrix.
planes in 3D, level sets, level curves, normal vector, point-to-plane distance, tangent plane, tangent line to surface, partial derivatives, linear approximation, quadratic form, paraboloid, elliptic paraboloid, hyperbolic paraboloid, saddle surface, discriminant test, surface graphs, MATH 215 Midterm 1, Calculus III, multivariable calculus, University of Michigan