A silly question about equations....

[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!

 

[berkeman]

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!

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?

 

[A.T.]

A=B

That the equation for the magnitudes.

A= -B

That the equation for the vectors.

 

[Mark44]

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.

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).

 

[ViolentCorpse]

That the equation for the magnitudes.

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 kirchhoffs 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?

 

[Mark44]

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

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 same problem has confused me in circuit analysis when applying kirchhoffs 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?

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 V_i = 6V, then V₀ would be -6V. The equation is V_i + V₀ = 0.

 

[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! :)