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If you square the vector $\vec v = 12.5 \, \text{m s}^{-1} \, \hat i$, what direction is $\vec v$? Similarly, if you had a quantity divided by a vector, $\frac{13 \, \text{kg}}{1.6 \, \text{m}, \, \hat i}$, what is the direction?

I came across this in physics, and I have been treating the unit vectors almost as units, because it works mathematically for me (except when there's two directions). I know this is incorrect, which is what inspired this question.

Also, is my notation for writing vectors correct?

Davis Rash
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    Multiplication and division are not defined in vector spaces. (There is scalar multiplication, and the inner product for inner product spaces, but those aren't what you're looking for. There is a vector multiplication in $\mathbb{R}^3$, the cross product, but this is just an accident. If you "square" a vector with that product, you get the zero vector.) – symplectomorphic Aug 21 '16 at 04:25
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    You say in your edit that you "came across this in physics." Came across what? Dividing by a vector? You definitely did not. You speak of unit vectors, so perhaps you are confusing dividing a vector by a scalar (which is just a form of scalar multiplication) with the other way around (which is not defined). – symplectomorphic Aug 21 '16 at 04:39
  • @symplectomorphic Well, a derived equation for the time of a projectile's motion can be $t = \frac{v_v - u_v}{g}$. $g$ is a vector, $-9.80 , \text{m s}^{-2} , \hat j$. In my usage of vectors as units (which I know they're not) the $\hat j$'s would cancel. – Davis Rash Aug 21 '16 at 04:43
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    no, $g$ is not a vector in that equation; it's a scalar denoting the magnitude of the acceleration vector. – symplectomorphic Aug 21 '16 at 04:51
  • @symplectomorphic Ah! Thank you! I cannot believe I didn't realize I've been giving directions to magnitudes so much. – Davis Rash Aug 21 '16 at 04:53
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    you're welcome. It's an extremely common error for high school students to confuse scalars with vectors, or to forget whether a letter is supposed to stand for a scalar or a vector. – symplectomorphic Aug 21 '16 at 04:57

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Your notation for vectors would go fine with most. However, you cannot multiply or divide by vectors; you can only multiply or divide a vector by a scalar. In your case, there is no meaning to "the square of 12.5 metres per second east" or "13 kilograms divided by 1.6 metres east", but "twice 12.5 metres per second east" does have a meaning: 25 metres per second east. Since the operations you want to do are not defined, there is no associated direction.

Intuitively, one way to see that a system of multiplication and division by vectors to yield vectors doesn't exist is that there are infinitely many unit vectors. This post has more on why division by vectors isn't defined.

There is one exception to this no-go principle: complex numbers are sometimes represented as vectors and they do admit multiplication and division. But since you put units in your quantities, they aren't complex numbers…

Parcly Taxel
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