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 Dot Product and the Law of Cosines

How can multiplying matching components and adding them up possibly know the angle between two arrows?

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Your summary note

1. Componentwise definition $\vec{A}\cdot\vec{B} = a_1b_1+a_2b_2+a_3b_3$
  Write the componentwise recipe, note that it extends to $n$ components, and record that the output is always a scalar and never a vector.
2. Geometric theorem $\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos\theta$
  State the theorem with $\theta$ the angle between the two vectors, and record that this one cheap computation carries both length and angle information.
3. The case $\vec{A}\cdot\vec{A} = |\vec{A}|^2$
  Carry out the first step of the proof by evaluating the geometric formula at $\theta = 0$ and matching it against $a_1^2+a_2^2+a_3^2$ .
4. Proof by comparison with the law of cosines
  Set $\vec{C} = \vec{A}-\vec{B}$ , write the law of cosines for $|\vec{C}|^2$ , expand $(\vec{A}-\vec{B})\cdot(\vec{A}-\vec{B})$ into four terms, compare, and record the equivalence of the two statements.
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