# A silly question about equations...

1. May 16, 2015

### ViolentCorpse

Hi,

I'm not sure if this belongs in the Maths section, so I'm sorry if I've made a mistake.

I'm having trouble understanding the meaning of the most basic sorts of equations.
For example, if we have two exactly equal but opposite force vectors A and B acting on a body, then by newton's law the algebraic sum of these forces must equal zero. If you ask me to write these words in maths, I would write it like this:
A - B = 0,
which implies that
A=B

But that goes against my intuition. I mean to say that, if you ask me to immediately express the relationship between two vectors that are equal yet opposite, in mathematical form, I would write it like this:

A= -B

I'm confused which one of these equations expresses the relationship correctly. Please help me understand what is the correct way of expressing this mathematically, and why.

Thank you!

2. May 16, 2015

### Staff: Mentor

The sum of the forces is equal to zero implies A + B = 0, not A - B = 0.

A and B would be force vectors, which have magnitude and direction. If A and B are equal and opposite, their magnitudes will be the same, and their directions will be opposite. Make sense?

3. May 16, 2015

### A.T.

That the equation for the magnitudes.

That the equation for the vectors.

4. May 16, 2015

### Staff: Mentor

They can't be "exactly equal" if their directions are different. As already pointed out, their magnitudes can be equal even though the vectors themselves are different (and therefore unequal).

5. May 17, 2015

### ViolentCorpse

But I found that equation out using Newton's law, which I think handles vectors?

The same problem has confused me in circuit analysis when applying kirchoff's voltage law. Suppose we have a battery of volts Vi connected to a resistor whose voltage is Vo. By KVL, the equation would be

Vi-Vo=0
Vi=Vo

Although Vi and Vo have opposite polarities. Again, I think Vi= -Vo, would be the correct expression of the relationship. Is the KVL equation only about magnitudes like Newton's?

6. May 17, 2015

### Staff: Mentor

Sure, but you're misunderstanding what the vectors represent. If they are oppositely directed, with equal magnitudes, then their vector sum will be the zero vector. For example, suppose $\vec{A}$ = <1,1> and $\vec{B}$ = <-1, -1>. The first vector points in the NE direction, and the second points in the SW direction, so their directions are directly opposite one another. Both vectors have magnitudes of $\sqrt{2}$.

$\vec{A} + \vec{B}$ = <1, 1> + <-1, -1> = <0, 0>.
The business about the opposite polarities is where the signs come in. If you take the voltage potential across the battery to be positive, then the voltage drop across the resistor should be treated as negative. Assuming Vi = 6V, then V0 would be -6V. The equation is Vi + V0 = 0.

7. May 17, 2015

### ViolentCorpse

So basically, I'm making the mistake of changing the sign + to - during the addition of symbols that represent the quantity, without plugging in the quantity first, right?

Thanks a lot guys! :)