Changes

I noticed at the end of calc3.6 that you used <math>\vec{0}</math>.  Does this mean a vector with 0 magnitude and direction?  Can a vector have direction without magnitude, for that matter? I'd never seen an actual number with the vector sign over it before. [[User:DouglasA|DouglasA]] 20:49, 11 February 2010 (EST)

I noticed at the end of calc3.6 that you used <math>\vec{0}</math>.  Does this mean a vector with 0 magnitude and direction?  Can a vector have direction without magnitude, for that matter? I'd never seen an actual number with the vector sign over it before. [[User:DouglasA|DouglasA]] 20:49, 11 February 2010 (EST)

:Yes, it's important to remember that when we talk about a vector of zero length, we don't mean just the number 0, which is a number, ie, a member of <math>\mathbb{R}</math>, we mean the ''vector'' <math>\begin{pmatrix}0\\0\\0\end{pmatrix}\in\mathbb{R}^3</math>.  Or, in the plane, we may mean <math>\vec{0}\in\mathbb{R}^2</math>.  Notably, we cannot add a number and a vector: <math>0+ \begin{pmatrix}x\\y\end{pmatrix}</math> is undefined, but <math>\vec{0}+\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}</math>.  We can't take the dot product, or cross product, of a number and a vector, but <math>\vec{0}\cdot\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}\times\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}</math>.  [[User:JacobB|JacobB]] 21:39, 11 February 2010 (EST)

:Yes, it's important to remember that when we talk about a vector of zero length, we don't mean just the number 0, which is a number, ie, a member of <math>\mathbb{R}</math>, we mean the ''vector'' <math>\begin{pmatrix}0\\0\\0\end{pmatrix}\in\mathbb{R}^3</math>.  Or, in the plane, we may mean <math>\vec{0}\in\mathbb{R}^2</math>.  Notably, we cannot add a number and a vector: <math>0+ \begin{pmatrix}x\\y\end{pmatrix}</math> is undefined, but <math>\vec{0}+\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}</math>.  We can't take the dot product, or cross product, of a number and a vector, but <math>\vec{0}\cdot\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}\times\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}</math>.  [[User:JacobB|JacobB]] 21:39, 11 February 2010 (EST)

::Hehe, I made a similar error to thinking <math>0=\vec{0}</math> above when I wrote <math>vec{0}\times\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}</math>.  Can't take the cross product of a two-vector! [[User:JacobB|JacobB]] 23:48, 11 February 2010 (EST)

::Hehe, I made a similar error to thinking <math>0=\vec{0}</math> above when I wrote <math>\vec{0}\times\begin{pmatrix}x\\y\end{pmatrix}=\vec{0}</math>.  Can't take the cross product of a two-vector! [[User:JacobB|JacobB]] 23:48, 11 February 2010 (EST)