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∞

Scaling and Adding Vectors: The Parallelogram Rule

How can laying two arrows head-to-tail and simply adding their components ever produce the very same arrow?

Loading…

←Previous Vectors, Components, and Length in Space Next The Dot Product and the Law of Cosines →

Your summary note

1. Scalar multiplication of a vector
  Record that multiplying by a scalar stretches, shrinks or reverses a vector, and show how $3\hat{i}+2\hat{j}+\hat{k}$ and $\langle 3,2,1\rangle$ describe the same arrow.
2. Vector addition and the parallelogram rule
  Draw the head-to-tail construction giving the diagonal $\vec{A}+\vec{B}$ , use it to state $\vec{A}+\vec{B}=\vec{B}+\vec{A}$ , and write the componentwise formula $\langle a_1+b_1, a_2+b_2, a_3+b_3\rangle$ .
3. Vector subtraction as the other diagonal
  State that $\vec{A}-\vec{B}$ means adding $-\vec{B}$ head-to-tail, and mark it as the second diagonal of the same parallelogram.
Attempt 1 of 2