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 Component of a Vector Along Any Unit Direction

Why does a single dot product give a vector's component along any direction, and how does that let a pendulum pick its own axes?

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Your summary note

1. Two forms of the dot product
  Write both the componentwise definition $\vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 + a_3 b_3$ and the geometric form $|\vec{A}||\vec{B}|\cos\theta$ , noting both give the same scalar.
2. Component of $\vec{A}$ along a unit vector $\hat{u}$
  Show the projection length is $|\vec{A}|\cos\theta$ , and carry through the step using $|\hat{u}|=1$ to reach $\vec{A} \cdot \hat{u}$ .
3. Coordinate axes as the special case
  Record that dotting with $\hat{i}$ , $\hat{j}$ , $\hat{k}$ recovers $a_1$ , $a_2$ , $a_3$ , and state that the same formula works for any direction.
4. Pendulum application
  Describe resolving the weight along a tangent $\hat{T}$ and normal $\hat{n}$ to the arc, identifying the tangential component with swinging and the normal component with tension, computed via dot products.
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