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 2x2 Determinant and the Area of a Parallelogram

How does rotating one vector by ninety degrees turn a stubborn sine into a cosine, and a triangle's area into a 2x2 determinant?

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Your summary note

1. Triangle area $\tfrac{1}{2}|\vec{A}||\vec{B}|\sin\theta$
  Note that any polygon splits into triangles, then write the base-times-height-over-two derivation giving $\tfrac{1}{2}|\vec{A}||\vec{B}|\sin\theta$ , and record that the sine blocks a direct dot product.
2. The 90-degree rotation trick
  Rotate $\vec{A}$ counterclockwise to $\vec{A}'$ , use $\theta' = \tfrac{\pi}{2}-\theta$ and equal lengths to get $\cos\theta' = \sin\theta$ , converting $|\vec{A}||\vec{B}|\sin\theta$ into $\vec{A}' \cdot \vec{B}$ .
3. Components of the rotated vector $\vec{A}' = \langle -a_2, a_1\rangle$
  Reproduce the box-flipping argument that gives $\vec{A}' = \langle -a_2, a_1\rangle$ for counterclockwise rotation, and note $\langle a_2, -a_1\rangle$ for clockwise.
4. The determinant $a_1 b_2 - a_2 b_1$
  Compute $\vec{A}' \cdot \vec{B} = a_1 b_2 - a_2 b_1$ , write it as the $2\times2$ determinant, and record it as $\pm$ the parallelogram area, halved for a triangle and absolute-valued for a true area.
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