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 3x3 Determinant and the Volume of a Box

Why is the minus sign in the middle of the 3x3 expansion not optional, and what does the handedness of space have to do with volume?

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Your summary note

1. Expansion along the first row
  Write out the expansion of the $3\times 3$ determinant along its first row as three $2\times 2$ minors with alternating signs, recording the delete-row-and-column rule that produces each minor and the six terms it yields.
2. The alternating signs and the middle minus
  Record that the signs alternate plus, minus, plus, note the same pattern reproduces the $2\times 2$ case, and state the two reasons the middle minus is required: the volume theorem and the handedness of space.
3. Volume theorem for the parallelepiped
  State $\det(\vec{A},\vec{B},\vec{C}) = \pm$ the volume of the parallelepiped spanned by $\vec{A}$ , $\vec{B}$ , $\vec{C}$ , and record what a parallelepiped is and why the sign is ambiguous.
Attempt 1 of 2