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∞

The Equation of a Plane Through Three Points

How can flattening a box and demanding a right angle turn out to be the very same equation for a plane through three points?

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Your summary note

1. The flattened parallelepiped test
  State that $P$ lies in the plane exactly when the box built on $\overrightarrow{P_1P}$ , $\overrightarrow{P_1P_2}$ and $\overrightarrow{P_1P_3}$ is squashed flat, and expand $\det(\overrightarrow{P_1P},\overrightarrow{P_1P_2},\overrightarrow{P_1P_3})=0$ into an equation in $x$ , $y$ , $z$ .
2. The normal vector condition $\overrightarrow{P_1P}\cdot\vec{N}=0$
  Define $\vec{N}$ as a vector perpendicular to the plane, state that $P$ lies in the plane exactly when $\overrightarrow{P_1P}\perp\vec{N}$ , and write that condition out in $x$ , $y$ , $z$ .
3. The normal vector from the cross product
  Record $\vec{N}=\overrightarrow{P_1P_2}\times\overrightarrow{P_1P_3}$ , note that the reversed order gives the opposite vector, and state that every nonzero multiple of $\vec{N}$ is again a normal vector.
4. The triple product identifying the two methods
  Write $\overrightarrow{P_1P}\cdot(\overrightarrow{P_1P_2}\times\overrightarrow{P_1P_3})$ , identify it as the triple product equal to the determinant, and record that the two quantities always agree and vanish exactly for $P$ in the plane.
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