At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
If \(m^2 + n^2 = 1\), \(p^2 + q^2 = 1\) and \(mp + nq = 0\) then what is \(mn + pq\)?
Meh - there's actually nothing elegant about it. The guy just made it look like that. But still, it looks really clever and let's see if you guys can come up with it.
I assume we will need to use trignometry to solve it?
That sounds like a good way to do it... in the system of real numbers.
nvm was working assuming mp+nq=1
Haha, I'm really sorry about that.
\(\sf m^2+n^2=1\) There is a trig identity: \(\sf cos^2(x)+sin^2(x)=1\) So then: \(\sf m= cos~(\alpha)\) \(\sf n=sin~(\alpha)\) Do the same for: \(\sf p^2+q^2=1\) \(\sf p= cos~(\beta)\) \(\sf q=sin~(\beta)\) Now, they gave us mp+nq=0 Substitute the values we got above into this second equation: \(\sf mp+nq=0\) \(\sf cos(\alpha)cos(\beta)+sin(\alpha)sin(\beta)=0\) And there is a trig identity: \(\sf cos(\alpha)cos(\beta)+sin(\alpha)sin(\beta)=cos(\alpha-\beta)\) so we can simplify that further down to: \(\sf cos(\alpha-\beta)=0\) I hope I'm going in the correct direction so far xD
Yes, of course. But don't forget that there's a huge world of things that aren't real numbers.
Nice! here is another alternative that looks kinda neat \[mn+pq=mn(p^2+q^2)+(m^2+n^2)pq =(mp+nq)(np+mq)=0\]
That's exactly the solution I was about to post. Great job.
Where did you find this?
copied it from quora, there are plenty of solutions there but i liked that one the most
Haha, Anders Kaseorg. Same.
He's just brilliant. I've been stalking him for a few days.
Oooh, interesting :o
looks somewhat similar to this https://en.wikipedia.org/wiki/Brahmagupta's_identity
Haha, yes. :)