Euclidean Vector Introduction

A vector is a quantity that has both a magnitude and a direction.
June 19, 2017—Ethan Truelove

Vector Basics

Let’s start by making a vector in the 2D Cartesian plane.
Draw the vector v starting at point (0,0) and ending at point (1,1):
In[]:=
Graphics[{Arrow[{{0,0},{1,1}}]},AxesTrue]
Out[]=
This vector describes a quantity that is pointed in the northeast direction. Its norm, or magnitude, is defined as
2
(
x
2
-
x
1
)
+
2
(
y
2
-
y
1
)
.
Calculate the norm of v, or Norm[v].
In[]:=
Sqrt[(1-0)^2+(1-0)^2]
Out[]=
2
Thus the magnitude of this vector v (||v||) can be said to be
2
. This vector v would be written as {1,1} because it moves over 1 unit in the x direction and up 1 unit in the y direction.

Unit Vectors

Unit vectors are vectors that have a direction, but have a magnitude of 1.
Show the unit vector i in blue and j in red:
In[]:=
Graphics[{Blue,Thick,Arrow[{{0,0},{1,0}}],Red,Arrow[{{0,0},{0,1}}]},AxesTrue]
Out[]=
The vector {1,1} could also be written as i + j, where both terms have a coefficient of 1. Unit vector i corresponds to the x component (horizontal), while j corresponds to the y component (vertical).

Vector Decomposition

Vector Addition and Subtraction

Vector Multiplication: Dot Product

Vectors in 3D

Vector Multiplication: Cross Product

FURTHER EXPLORATIONS
Parametric Equations of Vectors
Vector Fields
Tangent Vectors
AUTHORSHIP INFORMATION
Ethan Truelove
6/19/17