Say we have three forces $F_1, F_2, F_3$, such that

$$F_1 + F_2 - F_3 = 0\hspace 10mm (1)$$

And let us say that $F_1$ and $F_2$ have the same direction and magnitude, and that $F_3$ has double the magnitude of either, in the opposite direction.

From this it would seem that $F_1$ and $F_2$ had the same direction (in highschool physics, at least!), but if we treat these vectors like numbers, we can make another statement:

$$F_1 - F_3 = - F_2\hspace 10mm (2)$$

And yet this seems absurd to me, since equality of vectors seems to imply equality of direction. From the statement in $(1)$ I can also state that: $$F_1 = F_2\hspace 10mm (3)$$

But this contradicts $(2)$!

Edit: So they don't contradict, but I guess what I was wondering was the notation -- that is, if we say that there is a force $F = ma$, $F - ma = 0$ follows. Does this mean that $ma$ and $F$ are in opposite directions? What does the negative sign really mean?

I'm sure I've probably totally missed the point of vectors, but I can't seem to be able to contract this question into a Google search.

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How do you get (3) from (1)? It does not follow. There is no reason for $F_1$ and $F_2$ to be in the same direction, and equation (2) is perfectly fine. Does this help? –  Michael Brown Mar 26 '13 at 6:32
Your first assumption is wrong. $F_1 + F_2 - F_3=0$ never implies $F_1$ and $F_2$ are in the same direction. You can definitely add forces in different directions and the fact that their sum is zero means you can construct a triangle with these vectors (adjust the direction accordingly). Check this link: en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction –  Debangshu Mar 26 '13 at 6:35
if $F_1 = F_2$, it means $F_3 = 2F_1$. So, equation (2) is $F_1 -F_3 = F_1 - 2F_1 = -F_1$. Now, since $F_1 = F_2$, we can also write $-F_1 = -F_2$ which proves the second statement: $F_1-F_3 = -F_2$ and hence there is no contradiction between (2) and (3). –  Debangshu Mar 26 '13 at 6:47
@Soyuz I see where you are getting hung up. $-F$ has the opposite direction as $F$. From equation (1) $F_3$ is in the same direction as $F_1 + F_2$, not the opposite direction. –  Michael Brown Mar 26 '13 at 7:06
@user9886: It's not really a bad question, why should he remove it? –  Manishearth Mar 27 '13 at 6:00

So to address your 2nd question regarding $F = ma$ and its simple transformation through subtraction $F - ma = 0$. In this case $\vec F$ and $\vec a$ are vectors, $m$ is simply a scalar multiplier on $a$ that is along for the ride. When you subtract one vector from another this requires that you flip the direction of the vector in order to do the subtraction, thus if the two vectors are equal you'll get $\vec 0$ as the result.

Really this isn't an different than scalar subtraction if you have $x = y$, then $x-y = 0$.

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