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I've learned to write a vector on one line as for example $(a,b,c)$. This can get very confusing if each component of the vector is an expression of several terms like this: $(a+b,b+c,a+b)$ since the commas get mixed up in the mess of signs.

Adding more parentheses doesn't really help since it gets mixed up with the already written parentheses like $((a+b),(b+c),(a+c))$. It almost doesn't look confusing here, but with more complex expressions, it can look very messy.

What I usually do is to write something like $([a+b], [b+c], [a+c])$ and then add an explanation for what the newly introduced parentheses mean, but I'd rather use a standard notation that doesn't require an explanation.

What are the alternatives that can be used without needing extra explanation?

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    $\begingroup$ Some whitespace...? $\endgroup$ – Hans Lundmark May 16 '18 at 20:06
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    $\begingroup$ I think that $(a+b,b+c,a+b)$ is sufficiently clear, as an alternative I use also $(a+b\quad b+c \quad a+b)$. $\endgroup$ – user May 16 '18 at 20:08
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You can make the parentheses bigger and put white spaces before and after the commas (Hans' comment beat me to it). For example: $$\big( a^2+3b+c~,~b^2-4c+d^2~,~e^a + \sin b + \log c \big)$$is produced with \big( a^2+3b+c~,~b^2-4c+d^2~,~e^a + \sin b + \log c \big). Here ~ is an unbreakable space, and you can use \Big or \Bigg as needed.

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$\mathbf{v}=\left<a+b,\space b+c,\space c+a \right>=(a+b)\mathbf{i}+(b+c)\mathbf{j}+(c+a)\mathbf{k}$

The bold font helps distinguish individual terms for me

Sometimes I'll also write

$\mathbf{v}=\left<a+b\space|\space b+c\space|\space c+a\right>$

Just keeping in mind that the vertical bars are separators, not to be interpreted as $\operatorname{Abs}(b+c)$

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