Multivariable Calculus

Vectors and Dot Products

01Vectors, Components, and Length in Space 02Scaling and Adding Vectors: The Parallelogram Rule 03The Dot Product and the Law of Cosines 04Finding Angles in Space With the Dot Product 05The Perpendicularity Test and the Normal Vector to a Plane Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Determinants and the Cross Product

01The Component of a Vector Along Any Unit Direction 02The 2x2 Determinant and the Area of a Parallelogram 03The 3x3 Determinant and the Volume of a Box 04The Cross Product and the Right-Hand Rule 05The Triple Product and the Volume of a Parallelepiped Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Matrices and Inverse Matrices

01Why Swapping a Cross Product Flips Its Sign 02The Equation of a Plane Through Three Points 03Matrix Multiplication: Rows Dotted With Columns 04Matrices as Transformations: Identity and Rotation 05The Inverse Matrix and Solving AX = B 06Computing a 3x3 Inverse: Minors, Cofactors, Adjoint Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Planes and Square Systems

01Planes From a Point and a Normal Vector 02Reading the Normal Vector Off a Plane Equation 03Three Planes, One Point: Solving a 3x3 System 04Invertibility, det A, and the Homogeneous System 05When det A = 0: Coplanar Normals and a Line of Solutions Problem set0/10 Problem set 20/10 Practice∞

Three Planes, One Point: Solving a 3x3 System

When three planes fail to meet in a point, why can the solution set be a line or empty — but never exactly two points?

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Your summary note

1. The unique solution as a common point
  Write out the example system, record that the first two planes meet in a line, and that the third cuts that line at the point computed by $X = A^{-1}B$ .
2. No solution, a line, or a plane of solutions
  List the possible solution sets, record which position of the line $\mathcal{P}_1 \cap \mathcal{P}_2$ relative to $\mathcal{P}_3$ gives each, and reproduce the argument ruling out exactly two solutions.
3. The signatures $0 = 0$ and $1 = 2$
  Record which outcome of elimination goes with infinitely many solutions and which with none, and write the sample contradiction $x + z = 1$ against $x + z = 2$ .
Attempt 1 of 2