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In computer science, we would write a condtional statement like so:

if (cross_product(vec_a, vec_b) = 0) 
   parallel = true;

But if we wanted to represent something like this mathematically, then what would be the correct notation?

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    $\begingroup$ $a \times b = 0 \implies a \perp b$ $\endgroup$ – bubububub Jul 18 at 10:59
  • $\begingroup$ Well, it would be $\parallel $ for parallel rather than $\perp$ for perpendicular, but yeah, I get the point. Thanks. $\endgroup$ – Ryan Walter Jul 18 at 11:10
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    $\begingroup$ Yeah, I went with that one first, but got confused about the proper one to use. Good you fixed it yourself. $\endgroup$ – bubububub Jul 18 at 11:29
  • $\begingroup$ Your question is quite unclear. $\endgroup$ – Yves Daoust Jul 18 at 12:57
  • $\begingroup$ What is “something like this”? There is a thing called denotational semantics, which is a mathematical notation that describes exactly what your “if” statement does. But if you leave out the part where the value true is stored in a variable, for example, the denotation gets simpler. $\endgroup$ – David K Jul 18 at 15:38
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You would usually use implications and arrows for it, e.g.

cross_product(..) = 0 $\Rightarrow$ parallel = true

to say that one follows from the other.

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You could use an "if-then" statement. So, for example,

If $a \times b = 0$, then $a$ and $b$ are parallel.

But so far, this is still an incomplete mathematical statement, because we haven't explained what $a$ and $b$ are. So, a more complete statement would be

For every pair of vectors $a,b \in \Bbb{R}^3$, if $a \times b = 0$ then $a$ and $b$ are parallel.

There might be some ambiguity about what "parallel" means based on who you ask and how you define it (eg. is $(1,0,0)$ parallel to $(0,0,0)$?). So, to remove this potential ambiguity, we might prefer to use a more agreed upon terminology, which is that of linear dependence (two vectors are linearly dependent if and only if one of them is a scalar multiple of the other):

For every pair of vectors $a,b \in \Bbb{R}^3$, if $a \times b = 0$ then $a$ and $b$ are linearly dependent.


I prefer to write full English sentences as opposed to using logical symbols only. However, if you want to, you could say the following:

$\forall a,b \in \Bbb{R}^3,$ $a \times b = 0 \implies$ $a$ and $b$ are linearly dependent.

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