1. ## Vector Equality Proofs

I'm a bit confused about how to approach these questions. The only method that I have been using is trying to disprove options by counterexample to get to the solution.

Under what condition is: $\displaystyle |\vec A - \vec B| = A + B$

a) The magnitude of vector B is zero.

|<-2, 7> - <0, 0>| = <-2, 7> + <0, 0 >
sqrt(53) = sqrt(53)

b) Vectors A and B are in the same direction.

|<2, 0> - <3, 0>| = <2, 0> + <3, 0>
1 != 13

c) Vectors A and B are in perpendicular directions.

|<2, 3> - <-1, 2>| = <2, 3> + <-1, 2>
sqrt(10) != ~4.84

d) Vectors A and B are in opposite directions.

|<3, 0> - <-3, 0>| = <3, 0> + <-3, 0>
6 = 6

e) The statement is never true

I'm very confused as to how to come to a definitive answer since a, d, and e are still in the game.

And I have to do $\displaystyle |\vec A - \vec B| = A - B$ next. :/

2. ## Re: Vector Equality Proofs

If you have given examples in which |A+ B|= |A|+ |B| the anyone of those is a counter-example to (e).

But for (a), (b), (c), and (d) if you are trying to prove that under each of those conditions, |A+ B|= |A|+ |B|, an example is NOT a proof.

3. ## Re: Vector Equality Proofs

Originally Posted by Algebrah
I'm a bit confused about how to approach these questions. The only method that I have been using is trying to disprove options by counterexample to get to the solution.

Under what condition is: $\displaystyle |\vec A - \vec B| = A + B$
Sorry to tell you but the notation $\displaystyle |\vec A - \vec B| = A + B$ makes no sense.

$\displaystyle |\vec A - \vec B|$ is a nonnegative number (a scalar), the length of difference of two vectors.

I don't know what you mean by $\displaystyle A + B$. In what sense is that a nonnegative number?

4. ## Re: Vector Equality Proofs

I think that they want me to find the magnitudes of vectors A and B on the right side, then sum them to see if the scalar values are equivalent on each side. The problem is that I don't know how to do that without using specific chosen values.

I agree that the right side of the equation is confusing, but that was how it was written in the book.

5. ## Re: Vector Equality Proofs

I'll probably have to try to use constants b, c, etc. in order to prove each one.

6. ## Re: Vector Equality Proofs

Originally Posted by Algebrah
I think that they want me to find the magnitudes of vectors A and B on the right side, then sum them to see if the scalar values are equivalent on each side. The problem is that I don't know how to do that without using specific chosen values.

I agree that the right side of the equation is confusing, but that was how it was written in the book.
This is true: $\displaystyle \|\vec{A}-\vec{B}\|\le \|\vec{A}\|+\|-\vec{B}\|=\|\vec{A}\|+\|\vec{B}\|$.

But it is easy to find a counter-example to equality.

See if you can find such and post it here.

7. ## Re: Vector Equality Proofs

Originally Posted by Algebrah
Under what condition is: $\displaystyle |\vec A - \vec B| = A + B$
$\displaystyle |\vec A - \vec B| = A + B$ iff (1) $\displaystyle \vec{A}$ and $\displaystyle \vec{B}$ are collinear and have opposite directions, or (2) one of the vectors is null. Depending on the definition, (2) may be a special case of (1); then only (1) is necessary.

8. ## Re: Vector Equality Proofs

Meh, I got both of them wrong. I'm just plain stumped. Thanks for the help everyone.