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∞

Planes From a Point and a Normal Vector

Why does demanding a single right angle between a normal vector and a displacement give you the whole equation ax + by + cz = d?

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Your summary note

1. Two parallel planes from one normal vector $\vec{N}$
  Work both examples with $\vec{N} = \langle 1,5,10\rangle$ : the plane through the origin from $\overrightarrow{OP} \cdot \vec{N} = 0$ , and the plane through $P_0 = (2,1,-1)$ from $\overrightarrow{P_0P} \cdot \vec{N} = 0$ .
2. The coefficients $a$ , $b$ , $c$ as a normal vector
  Compare $x + 5y + 10z = 0$ with $x + 5y + 10z = -3$ , record that $a$ , $b$ , $c$ give $\vec{N}$ , and that only $d$ changes between parallel planes.
3. Getting $d$ by evaluating at a known point
  Record the shortcut for the constant, and check it on the example by substituting $P_0 = (2,1,-1)$ into $x + 5y + 10z$ to recover $-3$ .
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