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 Triple Product and the Volume of a Parallelepiped

Why does building a volume as base times height collapse into A dot (B cross C), the very determinant you were trying to avoid?

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Your summary note

1. Volume as base area times height
  Derive $\text{volume} = |\vec{B} \times \vec{C}|\left(\vec{A} \cdot \hat{n}\right)$ using base area $|\vec{B} \times \vec{C}|$ , height $\vec{A} \cdot \hat{n}$ , and $\hat{n} = (\vec{B} \times \vec{C})/|\vec{B} \times \vec{C}|$ , then cancel to get $\vec{A} \cdot (\vec{B} \times \vec{C})$ .
2. Order and parentheses in the triple product
  Record that the cross product must be computed first and note what goes wrong if $\vec{A} \cdot \vec{B}$ is done first.
3. Identity $\det(\vec{A}, \vec{B}, \vec{C}) = \vec{A} \cdot (\vec{B} \times \vec{C})$
  Write the first-row expansion of the determinant, write $\vec{A} \cdot (\vec{B} \times \vec{C})$ using the $i, j, k$ components of $\vec{B} \times \vec{C}$ as the three $2 \times 2$ minors, and match the two expressions term by term.
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