| | :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) |