Find u.v with Given Constraints: ||u+v||=1 and ||u-v||=5

[kuahji]

Find u.v given that ||u+v||=1 & ||u-v||=5.

The first thing I did was drawn a simple picture, it became apparent that u & v wouldn't be orthogonal. So then the Pythagorean Theorem wouldn't work. Next I moved on to squaring both sides
||u+v||$$^{2}$$=||u||$$^{2}$$+2(u.v)+||u||$$^{2}$$
However here again, I didn't seem to be getting anywhere because I can't do anything with the middle term. I also tried squaring the other equation, solving for ||u|| & substituting it into the other equation. But that still left me with two variables in one equations. So I'm kinda lost about what to actually do in this problem.

 

[Dick]

There isn't any unique solution. If you find a u and v that satisfy that then you can always rotate u and v by any angle and they will still satisfy that relation. Are you just supposed to find ANY u and v?

 

[kuahji]

Yes any solution. We had a similar one earlier that had infinitely many solutions. But it was specified that it was R^2 or R^3. Here it does not say. We haven't learned about rotations yet, that is the next section.

 

[Dick]

How about R^1? u=3, v=-2. Is that good enough? If it's R^2, u=(3,0), v=(-2,0). Etc.