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∞

Reading the Normal Vector Off a Plane Equation

How can one dot product decide whether a vector is parallel to a plane, perpendicular to it, or neither?

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Your summary note

1. The family of equations for one plane
  Write one plane equation and its doubled version side by side, record that both describe the same plane, and connect that to normal vectors of every length.
2. Perpendicular vectors as the scalar multiples of $\vec{N}$
  State that the vectors perpendicular to a plane are exactly the scalar multiples of $\vec{N}$ , then test $\vec{v} = \langle 1,2,-1\rangle$ against $\vec{N} = \langle 1,1,3\rangle$ for proportionality.
3. The parallelism test $\vec{v} \cdot \vec{N} = 0$
  Compute $\langle 1,2,-1\rangle \cdot \langle 1,1,3\rangle = 1 + 2 - 3 = 0$ and record the verdict it settles for $\vec{v}$ and the plane $x + y + 3z = 5$ .
4. The two readings of a zero left-hand side
  Record the distinction between them: the point $(1,2,-1)$ misses $x + y + 3z = 5$ but lies on the parallel plane $x + y + 3z = 0$ through the origin.
Attempt 1 of 2