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∞

Matrices as Transformations: Identity and Rotation

Why does the product AB apply B first, and which 2x2 matrix carries out a quarter turn of the plane?

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Your summary note

1. The product $AB$ as composition, $(AB)X = A(BX)$
  Write the associativity statement $(AB)X = A(BX)$ , and record that $AB$ applies $B$ first and then $A$ , exactly as $f(g(x))$ applies $A$ first.
2. Noncommutativity of the matrix product
  Record that $AB$ and $BA$ are different transformations, and state the size condition on widths and heights that can leave one of the two products undefined.
3. The identity matrix $I$ with $IX = X$
  Write the $3\times 3$ identity with ones on the diagonal and zeroes elsewhere, then carry out $IX$ entry by entry on $\langle x_1, x_2, x_3\rangle$ to recover $X$ .
4. Rotation by $90^\circ$ and the columns of $R$
  Write $R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ , compute $R\hat{i} = \hat{j}$ and $R\hat{j} = -\hat{i}$ , note the columns are those images, and carry $R^2$ through to $-I$ .
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