Automoderated users, edit
3,552
edits
Changes
→Vectors, and Vector Functions
=Vectors, and Vector Functions=
=Vectors, and Vector Functions=
==Vectors==
A '''vector''' is a mathematical object which has ''magnitude'' like a number, but also ''direction.'' For example, if I wanted to describe how a person is driving at any particular time, I would need two numbers: their speed, and some measure of their direction (the angle it makes with true north, for example). These two quantities together constitute the '''motion vector''' of that person.
A '''vector''' is a mathematical object which has ''magnitude'' like a number, but also ''direction.'' For example, if I wanted to describe how a person is driving at any particular time, I would need two numbers: their speed, and some measure of their direction (the angle it makes with true north, for example). These two quantities together constitute the '''motion vector''' of that person.
In three dimensional space we use three variables to write down a vector. In Cartesian coordinates, this would look like <math>\vec{v} = \begin{pmatrix}x\\y\\z\end{pmatrix} \ </math>.
In three dimensional space we use three variables to write down a vector. In Cartesian coordinates, this would look like <math>\vec{v} = \begin{pmatrix}x\\y\\z\end{pmatrix} \ </math>.
We can, if we wish, talk about just the length of the vector - this is its absolute value, or <math>\left\| \vec{v} \right\| \ </math>. For a vector written in Cartesian coordinates, the Pythagorean theorem tells us this is just <math>\sqrt{x^2 + y^2} \ </math>.
Vectors can be added and subtracted, just like numbers. See the illustration below.
In Cartesian coordinates, this is particularly easy: <math>\begin{pmatrix}x\\y\end{pmatrix} + \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x+a\\y+b\end{pmatrix} \ </math> and <math>\begin{pmatrix}x\\y\end{pmatrix} - \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}x-a\\y-b\end{pmatrix} \ </math>.
Multiplication by a number is also simple: the vectors direction is unchanged, but its length is multiplied by that number. Again, in Cartesian coordinates, this becomes simple: <math>c\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ca\\cb\end{pmatrix} \ </math>.
There are two ways to multiply a vector by another vector. In two dimensions, we can perform only one, which is called the '''dot product''' or in more advanced terminology, '''inner product.''' Given two vectors, this dot product is how far one vector goes in the direction of the other, using the other as the "unit" of length. It is defined <math>\vec{u} \cdot \vec{v} = \left\|\vec{u}\right\| \left\|\vec{v}\right\| \cos(\theta) \ </math>, where <math>\theta \ </math> is the angle between <math>\vec{u} \ </math> and <math>\vec{v} \ </math>. As before, this takes a particularly simple form in Cartesian coordinates: <math>\begin{pmatrix}x\\y\end{pmatrix} \cdot \begin{pmatrix}a\\b\end{pmatrix} =xa+by \ </math>. This expression is similar for the three dimensional case: <math>\begin{pmatrix}x\\y\\z\end{pmatrix} \cdot \begin{pmatrix}a\\b\\c\end{pmatrix} = xa+yb+zc \ </math>.
==Vector Functions==
Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, <math>f(x,y,z) = \ </math>.
=Coordinate Systems=
=Coordinate Systems=