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If I'm trying to write basic vectors, just as simple as the magnitude being 5 and the direction being zero, how would I do this? Would it be a row vector with parenthesis:$\overrightarrow{v} = (5, 0)$, a row vector with brackets: $\overrightarrow{v} = [5, 0]$, a column vector with parenthesis: $\overrightarrow{v} = \begin{pmatrix} 5\\ 0\\ \end{pmatrix}$, or a column vector with brackets: $\overrightarrow{v} = \begin{bmatrix} 5\\ 0\\ \end{bmatrix}$? Thank you if you can tell me what the correct notation for this simple vector is, everywhere I go seems to write them differently and the inconsistency makes me want to rip my hair out.

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    $\begingroup$ You can write it however you want, as long as you are consistent. $\endgroup$ – Morgan Rodgers Oct 14 '18 at 21:19
  • $\begingroup$ You might also want to consider what you are doing with these vectors. If you are frequently left-multiplying them by a matrix (e.g., $Av$), make them (thin) column vectors. If you are frequently right-multiplying them by a matrix (e.g., $vA$), make them (wide) row vectors. If you are performing both of these operations just as frequently, try not to mix row and column vectors too much (just pick one convention and apply transposes when necessary). As for the bracketing, just be consistent within one document (see Morgan Rodgers' comment). $\endgroup$ – parsiad Oct 14 '18 at 21:22
  • $\begingroup$ @MorganRodgers and parsia thank you, I'll take this into consideration! $\endgroup$ – jstowell Oct 14 '18 at 21:23
  • $\begingroup$ One thing I want to point out: the coordinates of a vector are not their magnitude and direction. The magnitude of the vector (3, 4) is 5, and the direction is about 53.13 degrees above the x-axis. $\endgroup$ – Deusovi Oct 16 '18 at 16:15
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So long as you make sure that you are consistent throughout the entire text, it's completely up to you. There are many different ways to represent vectors. In Linear Algebra, people like to use column notation, with either parentheses or square brackets, as this is the most convenient in linear algebra. Some people also write something like $\vec{v}=\begin{bmatrix}5&0\end{bmatrix}^T$ to have a column vector, but be able to write it nicely inline. In high school (perhaps up to freshman) level classes, the notation $\vec{v}=\langle 5,0\rangle$ is also used, but this ignores the fact that vectors are matrices with 1 row or 1 column, so is less widely used at a higher level. Ultimately it's your choice, whatever is most convenient to you, you should use it.

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All notation are used and correct depending upon the specific context and convention adopted.

Usually in linear algebra context vectors $\vec v$ are considered colummn vector and transponsed vectors $\vec v^T$ are row vectors that is

$$\overrightarrow{v} = \begin{pmatrix} 5\\ 0\\ \end{pmatrix} \quad \overrightarrow{v^T} = \begin{pmatrix} 5 &0\\ \end{pmatrix}$$

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As has been said in a comment, there are multiple ways. If you continue into mathematics you will see that in reality there is very little notation that is truly standard and people make new notation up all the time even though there was perfectly good notation before.

Vectors are particularly likely to have nonstandard notation because they pop up in virtually every field of mathematics, and the different fields can use them in completely different ways, so notations can be completely different. You may want to learn to live with it.

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