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∞

Matrix Multiplication: Rows Dotted With Columns

How does a table of numbers turn three linear formulas into the single equation AX = U, and why must rows meet columns for a product to exist?

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Your summary note

1. Linear relations packaged as $AX = U$
  Write a three-variable change of coordinates as explicit linear formulas, then record the same system as the matrix product $AX = U$ with $A$ , $X$ and $U$ identified.
2. The row-column dot product rule
  State that the entry in row $i$ , column $j$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$ , and compute the $14$ example.
3. Size of $AB$ and the compatibility condition
  Record that $AB$ has the height of $A$ and the width of $B$ , and that the product exists only when the width of $A$ equals the height of $B$ .
Attempt 1 of 2