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∞

When det A = 0: Coplanar Normals and a Line of Solutions

What does a flat, zero-volume box built from three normal vectors tell you about the solutions of the system they come from?

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←Previous Invertibility, det A, and the Homogeneous System

Your summary note

1. The rows of $A$ as the three normal vectors
  Record that the rows of $A$ are the normals, rewrite the condition as $\det(\vec{N}_1, \vec{N}_2, \vec{N}_3) = 0$ , and describe the flat box of volume zero they span.
2. The line perpendicular to the plane of the normals
  Reproduce the argument that the line through the origin perpendicular to the plane of the normals lies inside all three planes, and record the infinitely many solutions.
3. $\vec{N}_1 \times \vec{N}_2$ as a nontrivial solution
  Record that this cross product is perpendicular to $\vec{N}_3$ as well, and write it down as a solution pointing along the line of intersection.
4. The general system $AX = B$ and $\det A$
  Record the two cases: a nonzero determinant gives the unique $X = A^{-1}B$ , and a zero determinant gives none or infinitely many, never exactly one.
Attempt 1 of 2