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∞

Vectors, Components, and Length in Space

Why does measuring the length of an arrow in space take two right triangles stacked on top of each other rather than one?

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Your summary note

1. Vector: direction, magnitude, and components
  Define a vector as an arrow with direction and length, and write it both ways: $\vec{A} = \langle a_1, a_2, a_3 \rangle = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ using the unit vectors along the axes.
2. Scalars, notation, and no fixed starting point
  Record that length $|\vec{A}|$ is a scalar, that $\mathrm{dir}\,\vec{A}$ comes from dividing by the length, and that $\overrightarrow{PQ}$ moved elsewhere is the same vector.
3. Length formula $|\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
  Work through $\vec{A} = \langle 3, 2, 1 \rangle$ applying the Pythagorean theorem twice to get $\sqrt{14}$ , then state the general formula and note it works in any number of dimensions.
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