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

[thegreenlaser]

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?

 

[HallsofIvy]

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}:
$$\begin{bmatrix} x \\ y \\ z\end{bmatrix}$$.

You can see the code for that, or any LaTex, by double clicking on the expression.

 

[tiny-tim]

hi thegreenlaser!

xi +yj +zk is often easier to write,

and it's a lot easier to make cross-products with!

 

[Rap]

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.

 

[pmacias]

I understand the importance of using the most efficient and effective notation in different fields of study. In the case of vector notation, there is no one "best" notation that is universally preferred by all physicists, engineers, and mathematicians. Each notation has its own advantages and may be more suitable for different situations.

In linear algebra, matrix notation is commonly used because it allows for efficient and concise representation of multiple vectors and operations. This notation is particularly useful for solving systems of linear equations and performing transformations. However, it may not be as intuitive for representing geometric concepts.

On the other hand, the (xi +yj +zk) notation is more commonly used in physics and engineering because it directly relates to the Cartesian coordinate system and is more intuitive for representing geometric concepts. It is also easier to visualize and manipulate in three-dimensional space. However, it may not be as efficient for representing multiple vectors or performing matrix operations.

Ultimately, the choice of notation depends on the specific context and purpose of the vector representation. In physics, the (xi +yj +zk) notation may be more suitable for representing forces and motion in three-dimensional space, while in engineering, matrix notation may be more efficient for representing transformations and operations on multiple vectors. In mathematics, both notations may be used depending on the specific problem being solved.

In conclusion, there is no one "best" notation for vector representation in physics, engineering, and mathematics. It is important for scientists to be familiar with both notations and use them appropriately in different contexts to effectively communicate and solve problems.