#Linear algebra

255 messages · Page 1 of 1 (latest)

ripe stream
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Find the least distance between the ending point of a position vector r and the plane that is defined by three non colinear points with position vectors r1,r2,r3.
Can someone help? Cause I have no idea how to do this.

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jade plaza
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Hello @ripe stream, try visualizing it. Do a sketch of a plane and a vector outside of that plane. what would be the minimal distance.

ripe stream
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But how do I even describe a plane mathematically

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I would guess I have to find the distance between two points, the one at the end of the vector and one on the plane. But I can’t arbitrarily choose a random point

jade plaza
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How would you describe a plane using a single vector ?

ripe stream
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How could I describe it?

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Maybe a vector perpendicular to the plane?

jade plaza
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YES !

ripe stream
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Really?

jade plaza
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Yes

ripe stream
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Nice

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So I need a cross product I guess?

jade plaza
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Not just one cross product

ripe stream
jade plaza
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Yes !!

ripe stream
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Ok so I guess r2-r1 and r3-r2 or something

jade plaza
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Good

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Do you have some data or is it just an abstract question ?

ripe stream
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Abstract

jade plaza
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Ok

ripe stream
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So now I work out the cross product between these vectors

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And find a perpendicular vector n

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Which I assume I have to make unitary?

jade plaza
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Yeah

ripe stream
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So n/|n|

jade plaza
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But I suggest you make everything unitary

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And create an orthonormal base

ripe stream
jade plaza
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Yes

ripe stream
jade plaza
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Absolutely

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Cause as you said, you're gonna be doing orthogonal projections.

ripe stream
jade plaza
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So you wanna make your life simple

ripe stream
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So where do I go on from here?

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I’m stuck

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I mean how has this helped with the exercise though?

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I now have a perpendicular unitary vector

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But how does that help me find the distance between the vector r and the plane?

jade plaza
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Did you create you basis in the plane ? is it orthonormal ?

ripe stream
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Wait lemme do it

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Maybe it’ll help me

jade plaza
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Is this an exercice or are you just challenging yourself ?

ripe stream
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@jade plaza how do I find the magnitude of n?

jade plaza
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I'll be right back give me 5 (hungry)

ripe stream
jade plaza
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Can I draw something for you ?

ripe stream
jade plaza
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You're basically trying to find the green vector

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which is the difference between the big red vector and the small one

ripe stream
jade plaza
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Your original plane doesn't go through the origin point (not linear)

ripe stream
jade plaza
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Noooo

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It's parallel to your original plane, so that your basis works well

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Now what you need to do is find the 2 red vectors

ripe stream
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Just a plane parallel to the original?

jade plaza
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yes and that goes through the origin

ripe stream
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Alright

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So we basically consider a plane parallel to the original that goes through the origin

jade plaza
ripe stream
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How can I find the big red vector?

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And the green one?

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I have to project r somehow, I get that

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But how?

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I only have a vector perpendicular to the plane

jade plaza
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ok, so small red vector is r projected on your plane. That will be easy since you did your gram schmidt thing, so it's the normal component of r. That's clear ?

ripe stream
jade plaza
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Look at the small red vector ...

ripe stream
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If I find the projection of r I can find the red vector too

jade plaza
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It's normal to your planes right ?

ripe stream
jade plaza
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yes

ripe stream
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Yeah it is

jade plaza
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So if you express r in your beautifully orthonormal basis. the "normal" component of r is that vector

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the small red one

ripe stream
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Yeah

jade plaza
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I drew the basis in blue

ripe stream
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But first I have to make the basis

jade plaza
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yeah but you can make that easily with gramm schmitt as you said

ripe stream
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To use gramm Schmidt I have to have a basis already right?

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I need three vectors b1,b2,b3 that are already a basis of E3 and then I make an orthonormal basis out of them

jade plaza
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Well you have all your vectors expressed in a basis right ?

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You have their coordinates so that's in a basis

ripe stream
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No wait

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I’ll use n r2-r1 and r3-r2?

jade plaza
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yes

ripe stream
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Alright

jade plaza
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Anyway, whatever you take, your first vector needs to be n

ripe stream
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Well n=(r2-r1)*(r3-r2)

jade plaza
ripe stream
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I don’t have a cross product symbol sorry lol

jade plaza
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in onrder to obtain n you need to solve :
n * (r2-r1) = 0
n * (r3-r2) = 0

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That will give you n

ripe stream
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Ohhhh

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I didn’t think of that

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I only thought of using cross product

jade plaza
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cross product could work yeah

ripe stream
jade plaza
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Would, sorry, I'm being a diva

ripe stream
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No I mean like

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It would actually work?

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Cause I tried it and I don’t understand how to do it

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Would I use a determinant?

jade plaza
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It would, the cross product is always perpendicular to the original vectors

ripe stream
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Yeah but how do I compute it here

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Anyway imma try what you said

jade plaza
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you take n = (x, y, z) and you solve

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But what are you computing?? you told me this was all conceptual ?

ripe stream
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That’s why idk what to compute

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I would assume I have to express r1 r2 and r3 as through random coordinates

jade plaza
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Forget computing we can do examples later if you want, you can pick any set of vectors and try

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Let's finish the proof for now

ripe stream
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Yeah Okk sorry

jade plaza
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Ok, so small red vector clear ??

ripe stream
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Wait no I haven’t found n yet 😭

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Tbh idk how to make a basis

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Like

jade plaza
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wtf are you trying to find ?

ripe stream
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I know how it works with the formula

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But idk how to do it here

jade plaza
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😄 we're just conceptualizing

ripe stream
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Yeah I mean

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n * (r2-r1) = 0
n * (r3-r2) = 0

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What do I do with these?

jade plaza
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To determine n, there are 2 :

  • Either you use the cross product (like you said)
  • or you use 2 scalar products (like I said)

Both work

jade plaza
ripe stream
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Ok so now how do we go on😭

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I’m sorry I’m so confused with this exercise

jade plaza
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so where do you want to pick this up ?

ripe stream
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Ok let’s say we have the orthonormal basis in terms of the vectors n,r1-r2,r3-r2

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How exactly do I project r now?

jade plaza
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Lets call them n, u1, u2

ripe stream
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Okk

jade plaza
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n being the normal component to the planes

ripe stream
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Yep

jade plaza
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the small red vector is just the n-component of vector r right ?

ripe stream
jade plaza
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Sure ? understood ?

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look at the draft

ripe stream
jade plaza
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perfect

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now the big red vector. how do you think we can get it ?

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hint, it's also a normal component of a vector we know

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Look at the draft

ripe stream
jade plaza
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Yes, that’s why I put a minus, big red - small red

ripe stream
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Oh kkkk

ripe stream
jade plaza
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Let me give you a hint. It’s not just one vector

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There are 3 of them

ripe stream
jade plaza
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It could

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But just take any point on the plane and it would work

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so either r1, r2, r3

ripe stream
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Oh yeah that makes sense

jade plaza
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I added r1 and r2 to make it clear

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with the right basis, they will all have the same vertical component

ripe stream
jade plaza
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yes

ripe stream
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How can we express the n component of r?

jade plaza
ripe stream
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Isn’t it r-projection of r?

jade plaza
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yes but you can just express r (as a vector) in the new basis

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as you're gonna do with r1 btw ...

ripe stream
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Yeahh

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Ok

jade plaza
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There are other ways

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for example, if you don't want to go though the whole gramm schmidt, you can just focus on the normal vector and do projections

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you'd be projecting (r2-r) projected along n

ripe stream
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I don’t understand that

jade plaza
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So

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Small red = - r . n

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right

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?

ripe stream
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Yeah

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Wait

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Is that -r*n?

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Dot product?

jade plaza
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yes dot

ripe stream
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Omg

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Bruhhhhhhhhhhhh

jade plaza
ripe stream
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I feel so dumb rn

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Yeah ffs

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I couldn’t figure out how we find the red vectors

jade plaza
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But make sure to have a unitary vector n !!

ripe stream
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Yepp

jade plaza
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Yeah I guess gramm schmidt was a bit overkill

ripe stream
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So then I find the magnitude of the green vector and that’s it?

jade plaza
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you can just do projections

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YES

ripe stream
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But this works perfectly

jade plaza
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There's a 3rd way

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😄

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involving calculus

ripe stream
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Calculus?

jade plaza
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yes

ripe stream
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Huhh?

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Wait I’m curious now

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Tell me

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Pls

jade plaza
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So basically ... let me draw it

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So once you find the equation of the plane

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You have the coordinates of the point N (basically the vector r)

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You can express the coordinates of a generic point M belonging to the plane

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Then you can express vector MN, and distance MN²

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MN² is going to be a function of (x,y,z)

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You'd have to minimize that function meaning :

  • First derivative = 0
  • Second derivative > 0
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And solve

ripe stream
jade plaza
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M (x,y,z) belonging to the plane should verify :

n . (OM + r1) = 0

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OM (x,y,z), so that's gonna give you three equations ...

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Once you have those you're gonna want to minimize OM

ripe stream
jade plaza
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It's gonna be long if we do it now 😄

ripe stream
jade plaza
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Actually you gotta check if the formula is in your syllabus, cause if that's the case, we don't need all this hassle

ripe stream
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Cause this is a linear algebra class and we don’t even focus on planes so idek why this was an exercise to begin with

jade plaza
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I hope you understood well the first 2 proofs

ripe stream
jade plaza
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If I can give you an advice, when you feel stuck in algebra :

  • Either draw to visualize
  • or keep going with logical implications until you stumble on an insight
ripe stream
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I appreciate your help very much

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Tysm!!!!

jade plaza
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You’re very welcome. Have a good night

ripe stream
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