Vectors

Vectors

Vectors can be thought of as arrows in the plane, or rays of finite length. We add them together by first following one arrow, then another. Since we have defined a new mathematical object, we would like to know what properties it has.

If u, v, and w are vectors the following properties hold:

$v+w=w+v$

$u+(v+w)=(u+v)+w$

$v+0=0+v=v$

$v+(-v)=0$

Here we can think of the number zero as a vector by doing nothing.

Scaling Properties

Since vectors can be thought of as arrows, going from one point to another, we can make them longer while maintaining the direction they are pointing in.

If $\alpha ,\beta$ are non-zero real numbers, and v, w are vectors, we have the following properties:

$0*v=0~1*v=v$

$-1*v=-v$

$(\alpha +\beta )v=\alpha *v+\beta *v$

$\alpha (v+w)=\alpha *v+\alpha *w$

$(\alpha )\beta *v=\alpha (\beta *v)$

Magnitude

Since vectors are finite length arrows, we can compute the length of any vector by looking at a vector as a change in both the x direction and y-direction.

Given a vector v we define the magnitude of v, denoted $\Vert v\Vert$, to be the length of v.

Properties: If v is a vector and c is a scalar:

$\Vert v\Vert \geq 0$

$\Vert c*v\Vert =|c|*\Vert v\Vert$

$\Vert -v\Vert =\Vert v\Vert$

$\Vert v\Vert =0{\text{if and only if v = 0}}$

Computation with Vectors

Up until now only the magnitude we have only talked about vectors in an abstract manner. We will now describe how we do arithmatic with them.

Given two points (x, y), and (a, b), the vector from (x, y) to (a, b) is < a - x, b - y>. We also sometimes notate a vector as $(a-x){\textbf {i}}+(b-y){\textbf {j}}$

This means we can talk about vectors in terms of change in each coordinate. We say that two vectors $v=<a_{1},b_{1}>,~w=<a_{2},b_{2}>$ are equal if $a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}$

Now we will describe how to add, subtract, scale, and find the magnitude of a vector:

If $v=<a,b>,w=<x,y>$ are vectors, and c is a scalar,

$v+w=<a+x,b+y>=(a+x){\textbf {i}}+(b+y){\textbf {j}}$

$v-w=<a-x,b-y>=(a-x){\textbf {i}}+(b-y){\textbf {j}}$

$c*v=(c*a){\textbf {i}}+(c*b){\textbf {j}}$

$\Vert v\Vert ={\sqrt {a^{2}+b^{2}}}$

Unit Vectors

For any non-zero vector, v, there is a vector of length one pointing in the same direction as v. We call this vector the unit vector in the direction of v.

Given a vector v, the unit vector in the direction of v, called u, is given by

$u={\frac {v}{\Vert v\Vert }}$

Finding Vector Given Magnitude and Direction

It was mentioned earlier that a vector is an arrow of finite length pointing in a fixed direction. So given the magnitude and direction a vector is pointing in we can find the vector v.

Given a vector, v, has length $\Vert v\Vert$ with an angle of $\alpha$ from the positive x direction, we have that

$v=\Vert v\Vert (\cos(\alpha ){\textbf {i}}+\sin(\alpha ){\textbf {j}})=\Vert v\Vert <\cos(\alpha ),\sin(\alpha )>$

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