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∞

Why Swapping a Cross Product Flips Its Sign

Why does swapping the two vectors in a cross product cost a minus sign, and how does a determinant that is not really a determinant remember it?

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Your summary note

1. The symbolic $3\times 3$ determinant for $\vec{A}\times\vec{B}$
  Write the array with $\hat{i},\hat{j},\hat{k}$ in the top row, record that it is a memory device rather than a true determinant, and expand along that row into the componentwise formula.
2. Length as parallelogram area, direction by the right-hand rule
  State that $|\vec{A}\times\vec{B}|$ is the area of the parallelogram with sides $\vec{A}$ and $\vec{B}$ , and record the right-hand rule that picks one of the two perpendicular directions.
3. Anti-commutativity $\vec{A}\times\vec{B}=-\,\vec{B}\times\vec{A}$
  Write the identity, reproduce the right-hand-rule argument with the two vectors exchanged, and check the sign on the $\hat{i}$ component $a_2b_3-a_3b_2$ .
4. The special case $\vec{A}\times\vec{A}=\vec{0}$
  Record the result with both arguments: the componentwise formula evaluated on $\vec{A}$ twice, and the flat parallelogram spanned by $\vec{A}$ with itself.
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