Can someone give an example of abc that works? I dont think its possible in reals

[wanton void]

Hello @bold jungle!

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[golden prawn]

If A'A = I, then |A| = ±1

So 3abc-(a³+b³+c³)=±1
3abc = 2±1
abc = 3/3, 1/3 = 1,1/3

So the values are 1 and 1/3

[bold jungle]

Indiviual values for abc

Since a²+b²+c²=1

But a³+b³+c³=2???

[golden prawn]

Where did you get this from?

[bold jungle]

Orthogonal

Its a rotation matrix that means the basis vectors must have a magnitude of 1

[golden prawn]

It's not a rotation matrix though

Not all orthogonal matrices are rotation matrices

[bold jungle]

Ohhh

I didnt know that

[golden prawn]

Rotation matrix in 3D is weird.

I don't remember it

[bold jungle]

I thought all orthogonal matrices were rotational ones

So that holds in 2d but not in 3d

[golden prawn]

No? There are non-rotation orthogonal matrices in 2D too I'm pretty sure

[bold jungle]

Your method is a lot faster idk why they did all that but it does hold there

Hmm i do think all orthogonal matrices are rotational tho im like 99% sure

I had been using it in a lot of questions

If the det of a is +1 than its rotational according to wikj

So i guess i had been getting lucky xd

Det with -1 are reflections

Interesting

Gonna look into the proofs for tgat

Misconception aside the problem still stands that on regular multiplication we end up with sum of squares as 1

[golden prawn]

Ok yeah in 2d they have to be

[golden prawn]

Eh, don't do it like that

Just work it out, it's gonna be simpler

Det -1 means there's a reflection, but there could also be a rotation+reflection

[spring goblet]

I was about to ask this question in a different thread but but how can we be sure all orthogonal matrices in 3-D are not always rotational in nature?

[golden prawn]

Just a moment.

Nvm all orthogonal matrices have to be rotations

[bold jungle]

so det cannot be negative in 2 dimensions?

[golden prawn]

Because they're just a change of basis

[bold jungle]

now im confused 😕

[golden prawn]

Orthogonal matrices are matrices whose column vectors form an orthonormal basis of the space.

[bold jungle]

yes thats what i knew

[golden prawn]

So any orthogonal matrix would just be a matrix that changes the directions of coordinates you're working in, and since it's normalised, there's no scaling.

It's essentially a rotation of the axes

Man, I'm being dumb a lot of times today

[bold jungle]

but what about those with det -1 also that doesnt make sense because then it would a times a transpose = -I

ok

[golden prawn]

Those are not orthogonal by definition

[spring goblet]

That is the case where the right-handed coordinate becomes left-handed.

[bold jungle]

ahhhhhh

so they are still perpendicular

[golden prawn]

Orthonormal, yes.

Orthogonal and normalised

[bold jungle]

@

@golden prawn

so back to original q

[golden prawn]

Yes?

[bold jungle]

how is it possible

that a,b,c are real

[golden prawn]

Lemme see

Tbf it's a Mains question, so don't expect too much though

[bold jungle]

yeah i guess i was just bewildered

maybe im pissed because i didnt of the identity first

[golden prawn]

You good?

[summer swallow]

+solved @golden prawn