$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

2  Vectors

(Relevant Section of the Textbook: 12.2 Vectors)

We have talked about one fundamental use of \(n\)-tuples of real numbers: describing posititions in space. But they also play a foundational role in the theory of vectors, which help us measure not locations but directions.

Definition 2.1 (Vector) A vector is a directed line segment, an object that stores both a length (called its magnitude), and a direction. You may draw a vector as a directed line segment, or a little arrow in space.

Example 2.1 (Points vs Vectors) Which of the following quantities are positions? Which are vectors?

  • Where I parked my car.
  • The wind hitting me in the face
  • The location of mars in the solar system.
  • The velocity of mars in the solar system.
  • Gravity’s acceleration

Exercise 2.1 Come up with some of your own scenarios that are measured using: 2-d points. 3-d vectors. 4d points, 4d vectors.

To work with vectors, we need a means of writing them down using numbers. One idea is to use the same Cartesian coordinate system discussed in the previous chapter, to express a vector in components.

Definition 2.2 (Vector Notation) To help avoid confusing points and vectors, we will try to use different notations for the two. For a point, we use a simple letter, like \(p\) whereas for a vector we either make it bold like \(\mathbf{u}\) or decorate it with an arrow, \(\vec{u}\).

When using cartesian coordinates, we write a point inline with round brackets, \(p=(1,2,3)\) whereas for a vector we use angle brackets, \(\vec{u}=\langle 1,2,3\rangle\). We also sometimes write a vector as a column of numbers, instead of a row to save space, or for other stylistic reasons: in this case we just use round brackets for ease of typesetting. \[\vec{u}=\pmat{1\\ 2 \\3}\]

Definition 2.3 (The Zero Vector) The zero vector in an \(n\) dimensional space is the vector with no magnitude and no length. In coordinates, this is the \(n\)-tuple of all zeroes: \(\langle 0,0,0\rangle\)

2.1 Arithmetic of Vectors

Definition 2.4 (Vector Addition) Vector addition is defined to give the combined effect of two vectors: if \(\vec{u}\) and \(\vec{v}\) are vectors, \(\vec{u}+\vec{v}\) is defined geometrically by the diagonal of the parallelogram with sides \(\vec{u}\) and \(\vec{v}\). This is equivalent to the vector formed by stacking \(\vec{u}\) and \(\vec{v}\) on one another head-to-tail, in either order.

In cartesian coordinates, this is \[\pmat{x\\ y\\ z}+\pmat{a\\ b\\ c}=\pmat{x+a\\ y+b\\ z+c}\]


Vector addition can be used to describe complicated motion in terms of simpler pieces. Indeed, this idea was used by the ancient greeks in their planetary model, where the complicated motion of objects in the heavens was modeled as a combination of various circular motions (or epicycles). Below is an animation showing their model of Mars’ motion about the earth, decomposed in terms of a sum of three circles.

Definition 2.5 (Scalar Multiplication) If \(\vec{u}\) is a vector and \(c\) is a number, the vector \(c\vec{u}\) is defined to be the vector pointing in the same direction as \(\vec{u}\), but \(c\) times as long. In Cartesian coordinates, \[c\langle u_1,u_2,u_3\rangle = \langle cu_1,cu_2,cu_3\rangle\]

Because we think of real numbers as being the kind of things that can scale vectors, we often call them scalars.

Definition 2.6 (Linear Combination) A linear combination of a list of vectors is a new vector made by taking a sum of scalar multiples of the original list.

For example, if \(\vec{u}=\langle 1,2\rangle\) and \(\vec{v}=\langle 3,4\rangle\), then the following is a linear combination:

\[\vec{w}=7\pmat{1\\2}-2\pmat{3\\4}=\pmat{7-6 \\ 14-8}=\pmat{1,6}\]

In this calculation you saw a bit of vector arithmetic: it works just like the arithmetic of numbers, one coordinate at a time.

Theorem 2.1 (Vector Arithmetic) Let \(\vec{u},\vec{v}\) and \(\vec{w}\) be vectors, and \(c,k\) be scalars. Then:

\[\vec{u}+\vec{v}=\vec{v}+\vec{u} \hspace{1cm} \vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}\]




2.2 Coordinate Bases

Cartesian coordinates are built from a collection of perpendicular axes. Each of these axes has a direction that we call a standard basis direction

Definition 2.7 (Standard Basis) For \(\RR^2\) the standard basis vectors are the vectors \(\langle 1,0\rangle\) and \(\langle 0,1\rangle\), pointing along the positive direction of the \(x\) and \(y\) axes.

For \(\RR^3\), the standard basis vectors are teh vectors \(\langle 1,0,0\), \(\langle 0,1,0\rangle\) and \(\langle 0,0,1\rangle\) pointing along the positive direction of the \(x,y\) and \(z\) axes respectively.

In general \(n\)-dimensional space, the \(n\) basis vectors are the vecctors which have all \(0\)s as coordinates except a single \(1\). The vector whose \(1\) is in the \(i^{th}\) coordinate is called the \(i^{th}\) basis vector.

For example, \(\langle 0,0,0,1,0,0,0,0,0,0,0,0\rangle\) is the \(4^{th}\) standard basis vector of 12-dimensional space. In two and three dimensions we give each of the bases a unique letter to aid in readability, instead of dealing with messy subscritps for only a handful of symbols.

Definition 2.8 (Standard Basis in \(\RR^2\) and \(\RR^3\)) In \(\RR^2\) we write the standard basis vectors as \[\ihat =\langle 1,0\rangle\hspace{1cm}\jhat=\langle 0,1\rangle\]

In \(\RR^3\) we continue this, writing \[\ihat=\langle 1,0,0\rangle\hspace{1cm}\jhat=\langle 0,1,0\rangle\hspace{1cm}\khat = \langle 0,0,1\rangle\]

We can use these standard basis vectors to express any vector in space. For example, in \(\RR^3\) every vector is some amount in the \(\ihat\) direction, some amount in the \(\jhat\) direction, and some amount in the \(\khat\) direction. This means we can write any vector as a linear combination of these:


2.3 Magnitude and Direction Information

One common use for vectors is to give directions to get from one point to another: that is, given points \(p,q\) in space, we want a vector starting at \(p\) and ending at \(q\).


This vector encodes the magnitude and direction information of “if you are at \(p\) and you walk this amount in this direction, you’ll arrive at \(q\)”. We can construct such a vector

Definition 2.9 (Vector from Two Points) The displacement vector from a point \(p=(p_1,p_2,p_3)\) to a point \(q=(q_1,q_2,q_3)\) is

\[\vec{d}=q-p=\pmat{q_1-p_1\\ q_2-p_2\\ q_3-p_3}\] And, analogously in other dimensions.

Thus, the vector \(\langle x,y,z\rangle\) is the displacement vector from the origin$ to the point* \((x,y,z)\). Its length (or magnitude) is just the distance from the origin to its other endpoint, whih we know from the pythagorean theorem.

Definition 2.10 (Magnitude) The magnitude of the vector \(\vec{u}=\langle u_1,u_2,u_3\rangle\) is given by the pythagorean theorem: \[\|\vec{u}\|=\sqrt{u_1^2+u_2^2+u_3^2}\] And, analogously in other dimensions.

A vector of length 1 is called a unit vector. We think of these as measuring purely direction just as we think of numbers as measuring purely length.

Definition 2.11 (Unit Vector in Given Direction) If \(\vec{v}\) is a nonzero vector, the unit vector in direction \(\vec{v}\) is denoted \(\hat{v}\), and is calculated by dividing \(\vec{v}\) by its own magnitude: \[\hat{v}:=\tfrac{1}{\|\vec{v}\|}\vec{v}\]

Exercise 2.2 (Unit Vectors in Given Directions) Find a unit vector in the direction of \(\langle 1,2,3,4\rangle\).

Exercise 2.3 (Unit Vectors in Given Directions) Find a vector of length \(2\) in the direction of \(\langle 1,1,1,1,1,1,1,1\rangle\). Hint: find a unit vector in this direction. What happens to its length if you scalar multiply it by \(2\)?

2.4 Videos

2.4.1 Video Tutorial Series

Here’s a short series of videos going through the different vector properties discussed above: