Which Vector Notation is Best for Physics, Engineering, and Mathematics?

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thegreenlaser
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I'm curious, which vector notation is preferred by physicists/engineers/mathematicians? In linear algebra we used matrix notation exclusively, putting the x,y,z,... components down a column matrix. (no idea how to put this in latex). In all my other courses though, we've been using (xi +yj +zk) notation where x,y,z are the components of the vector and i,j,k are unit vectors on the x, y, and z axes respectively.

Which notation do you prefer for which situations and why?
 
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If I am writing quickly, just <x, y, z> will do. Typically, I would use the "column matrix" form only if I were working with matrices.

By the way, you can do matrices in LaTex with \begin{bmatrix}... \end{bmatrix} for "square brackets" or \begin{pmatrix} ... \end{pmatrix} for "parentheses". Use & to separate items on a single line and \\ to separate lines a single column matrix would be
\begin{bmatrix} x \\ y \\ z\end{bmatrix}:
[tex]\begin{bmatrix} x \\ y \\ z\end{bmatrix}[/tex].

You can see the code for that, or any LaTex, by double clicking on the expression.
 
It definitely depends on the application. The thing about vectors and tensors in physics is that they don't depend on the coordinate system you use to describe them. You can use coordinate free notation, like C=A+B, but it can get messy, its much easier sometimes to use the "language" of a coordinate system to talk about vectors, like C1=A1+B1, C2=A2+B2, etc. But then sometimes you have to deal with the fact that the vector and tensor equations using these coordinate systems are independent of those coordinate systems. This can get messy too, but Einstein developed a way of describing vectors using coordinate systems along with a bunch of rules about how to manipulate them which automatically shows you the invariance of the equations. Check out "Einstein notation" and "Coordinate free notation" on Wikipedia. It takes some work to get the hang of it, but once you do, its a very valuable tool in your vector/tensor toolkit.