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 Cross Product and the Right-Hand Rule

How can a determinant with i, j, k in its top row produce an honest vector whose length is an area and whose direction your right hand picks?

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Your summary note

1. Definition of $\vec{A} \times \vec{B}$ as a symbolic determinant
  Write the $3\times 3$ determinant with $\hat{i},\hat{j},\hat{k}$ in the first row and the components of $\vec{A}$ , $\vec{B}$ below, then expand along the first row into the vector formula with the minus sign on the $\hat{j}$ term.
2. Cross product versus dot product
  Record that the cross product outputs a vector while the dot product outputs a number, and note the symbolic determinant takes vectors rather than numbers as a mnemonic device.
3. Length $|\vec{A} \times \vec{B}|$ as parallelogram area
  State that the length equals the area of the parallelogram spanned by $\vec{A}$ and $\vec{B}$ , found by squaring each component, summing, and taking the square root, with no $\pm$ needed.
4. Direction by the right-hand rule and $\hat{i} \times \hat{j} = \hat{k}$
  State the direction is normal to the plane fixed by pointing along $\vec{A}$ , curling toward $\vec{B}$ , thumb out, and verify $\hat{i} \times \hat{j} = \hat{k}$ both geometrically and by the determinant.
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