Math Contest #25 [2 SBI]

in #puzzle2 years ago (edited)

Here you can keep your brain fit by solving math related problems and also earn SBI or sometimes other rewards by doing so.
The problems usually contain a mathematical equation that in my opinion is fun to solve or has an interesting solution.
I will also only choose problems that can be solved without additional tools(at least not if you can calculate basic stuff in your head), so don't grab your calculator, you won't need it.
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Rules

No upvote, No resteem, No follow required!

I will give the SBI(s) randomly to any participants.

You have 4 days to solve it.

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Problem

Given two vectors (a, 5a, 2a/x) and (3a, 4a, x).
For what values of a and x are the two vectors orthogonal to each other?

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To everyone who already participated in a past contest, come back today and try a new problem(tell me if you don't want to be tagged):
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In case no one gets a result(which I doubt), I will give away the prize to anyone who comments.

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@contrabourdon sponsors my contests with 2 STEEM weekly.
You can support him by using a witness vote on untersatz, so he can further support this and other contests.

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I read this only now...
Anyway in order to be orthogonal the scalar product must be zero.
The scalar product is 23a^2 + 2a , so "a" has to be -2/23 and x is free. But x has to be different from 0, otherwise the first vector is not defined. Another solution is a = 0 and x different from 0 but we get a null vector for the first one and it is not so much interesting.
Bye

I am stumped... but I will go with 0

The vector (0, 0, 0) cannot be orthogonal to another vector because it has no direction.

in reality (0,0,0) is orthogonal to every vector, since the scalar product between him and any vectors is always 0
;)

You are right. It depends how you define it.
The definition I use is that two vectors are orthogonal if they form a right angle.

Ok that's fine.
This concept is not at all trivial.
The physicists say that a vector is an object that has 3 features: length, direction and verse.
So if (0,0,0) is a vector must have one direction!!
But rightly as you say it has no direction.
On the other side the mathematicians define a vector as an object that "live" in a space with a certain number of dimensions: in this case 3.
So (0,0,0) is a vector respect to this point of view.
Moreover it's true that if two vector form a right angle, their scalar product is 0, and this is true not by definition but it is a fact. You can check it by using whatever method you prefer...arctg,Pythagorean theorem,...
So I think that the null vector is an extension of the physical vector concept, like I'm sure you know it happens for the elements of an Hilbert space on quantum physics, where the L2 complex functions are the vectors and the scalar product is an integral!!
That's really cool!!