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∞

Finding Angles in Space With the Dot Product

How do one dot product and two lengths pin down an angle in space, and what does the sign alone already reveal?

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Your summary note

1. Angle formula $\cos\theta = \frac{\vec{A}\cdot\vec{B}}{|\vec{A}|\,|\vec{B}|}$
  Rearrange the dot product theorem to isolate the cosine, and note that every quantity on the right is computable from components.
2. Worked example: angle at $P$ in a triangle
  Using $P=(1,0,0)$ , $Q=(0,1,0)$ , $R=(0,0,2)$ , build $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ , compute their lengths and dot product, then take the inverse cosine to get $\theta\approx 71.5°$ .
3. Sign of the dot product classifies the angle
  Record that positive means acute, zero means perpendicular, and negative means obtuse, since the positive lengths make the dot product's sign match the sign of $\cos\theta$ .
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