vector math/cross product
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I really don't want to ask this question here, but since there's no math forum I haven't much of a choice. Anyway, I have a math question for the gurus again. I'm trying to understand just how a cross product works rather than just do as I'm told kinda thing. So, given this...
| x1 | | x2 | | y1z2 - z1y2 |
| y1 | x | y2 | = | z1x2 - x1z2 |
| z1 | | z2 | | x1y2 - y1x2 |My question is, why is that so? I realize (using the dot product as a reference) that the elements are independent or so I thought. I would've thought that multiplying the two would mean somethign of this nature...
x1 * x2 + y1 * y2, etc.
But I realize that would just be the dot product again. Also, why do I need to subtract at all when multiplying? Can anyone please explain this to me? The book I'm reading did a great job at explaining the dot product, but not the cross product. TIA Jeremy Falcon
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I really don't want to ask this question here, but since there's no math forum I haven't much of a choice. Anyway, I have a math question for the gurus again. I'm trying to understand just how a cross product works rather than just do as I'm told kinda thing. So, given this...
| x1 | | x2 | | y1z2 - z1y2 |
| y1 | x | y2 | = | z1x2 - x1z2 |
| z1 | | z2 | | x1y2 - y1x2 |My question is, why is that so? I realize (using the dot product as a reference) that the elements are independent or so I thought. I would've thought that multiplying the two would mean somethign of this nature...
x1 * x2 + y1 * y2, etc.
But I realize that would just be the dot product again. Also, why do I need to subtract at all when multiplying? Can anyone please explain this to me? The book I'm reading did a great job at explaining the dot product, but not the cross product. TIA Jeremy Falcon
You're thinking too linearly. Start with a 2D cross product and draw both vectors and the resulting cross product vector on paper. You should have an appropriate "ah ha" moment. Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
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You're thinking too linearly. Start with a 2D cross product and draw both vectors and the resulting cross product vector on paper. You should have an appropriate "ah ha" moment. Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
Now I wish I had graph paper. Lemme go draw some out and see what happens. Jeremy Falcon
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You're thinking too linearly. Start with a 2D cross product and draw both vectors and the resulting cross product vector on paper. You should have an appropriate "ah ha" moment. Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
I think I get it. What I noticed so far is it seems to be for finding right angles (based off the first vector only/not cummutative) among two 3D vevtors vectors [edit] my typing has actually gotten worse :doh: [/edit]. Thanks for the insight. [edit²] Nope I was wrong. After playing around it's about finding a perpendicular vector for both x and y vectors. Step two is finding out how the hell I'm gonna use this I guess. [/edit²] One of these days I'll be an old hand at this stuff. :laugh: Jeremy Falcon
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Now I wish I had graph paper. Lemme go draw some out and see what happens. Jeremy Falcon
Jeremy Falcon wrote:
Now I wish I had graph paper.
Huh. All this technology, and I've never seen virtual graph paper. An ideal, simple, yet powerful application. Extensible. Hmmm.... Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
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Jeremy Falcon wrote:
Now I wish I had graph paper.
Huh. All this technology, and I've never seen virtual graph paper. An ideal, simple, yet powerful application. Extensible. Hmmm.... Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
And much easier to see in 3D. Come to think of it, it would make a nice starter OGL project and probably one that I would use. Of course, if you have the itchin' by all means. :laugh: Jeremy Falcon
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I really don't want to ask this question here, but since there's no math forum I haven't much of a choice. Anyway, I have a math question for the gurus again. I'm trying to understand just how a cross product works rather than just do as I'm told kinda thing. So, given this...
| x1 | | x2 | | y1z2 - z1y2 |
| y1 | x | y2 | = | z1x2 - x1z2 |
| z1 | | z2 | | x1y2 - y1x2 |My question is, why is that so? I realize (using the dot product as a reference) that the elements are independent or so I thought. I would've thought that multiplying the two would mean somethign of this nature...
x1 * x2 + y1 * y2, etc.
But I realize that would just be the dot product again. Also, why do I need to subtract at all when multiplying? Can anyone please explain this to me? The book I'm reading did a great job at explaining the dot product, but not the cross product. TIA Jeremy Falcon
Ah, where to start. The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling with a length equal to the area of the parallelogram defined by the vectors. (There are actually 2 vectors that meet this criteria, by default we choose one of them) The cross product only works in 3 dimensions. In 2D there isn't a perpendicular direction, and in higher then 3 dimensions there is more then just one perpendicular direction. For all these cases it makes more sense to use a bi-vector product which represents the plane directly instead of trying to map it to a vector.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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Ah, where to start. The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling with a length equal to the area of the parallelogram defined by the vectors. (There are actually 2 vectors that meet this criteria, by default we choose one of them) The cross product only works in 3 dimensions. In 2D there isn't a perpendicular direction, and in higher then 3 dimensions there is more then just one perpendicular direction. For all these cases it makes more sense to use a bi-vector product which represents the plane directly instead of trying to map it to a vector.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
Andy Brummer wrote:
The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling
Yeah I just got to that. It's amazing what graph paper can do. :-D Ok, you sound like you know this stuff, so what is the cross product used for in practical terms? The dot product I can envision being used when transforming objects easy enough, but is there some things useful for this for the direction I'm headed in (OGL programming)? Thanks for the reply! Jeremy Falcon
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Andy Brummer wrote:
The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling
Yeah I just got to that. It's amazing what graph paper can do. :-D Ok, you sound like you know this stuff, so what is the cross product used for in practical terms? The dot product I can envision being used when transforming objects easy enough, but is there some things useful for this for the direction I'm headed in (OGL programming)? Thanks for the reply! Jeremy Falcon
Jeremy Falcon wrote:
what is the cross product used for in practical terms?
Consider this: you can take two vectors on a surface, that'll tell you what plane the surface resides on. Find the cross product of those, and you can decide whether or not the surface is facing the "camera"... if it's not, don't bother drawing it. Handy, eh? (you can also use it for calculating lighting / shading, or drawing nifty arrows sticking out of a surface... if you're into that sort of thing.)
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I really don't want to ask this question here, but since there's no math forum I haven't much of a choice. Anyway, I have a math question for the gurus again. I'm trying to understand just how a cross product works rather than just do as I'm told kinda thing. So, given this...
| x1 | | x2 | | y1z2 - z1y2 |
| y1 | x | y2 | = | z1x2 - x1z2 |
| z1 | | z2 | | x1y2 - y1x2 |My question is, why is that so? I realize (using the dot product as a reference) that the elements are independent or so I thought. I would've thought that multiplying the two would mean somethign of this nature...
x1 * x2 + y1 * y2, etc.
But I realize that would just be the dot product again. Also, why do I need to subtract at all when multiplying? Can anyone please explain this to me? The book I'm reading did a great job at explaining the dot product, but not the cross product. TIA Jeremy Falcon
http://mathworld.wolfram.com/CrossProduct.html[^] Todd Smith
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Andy Brummer wrote:
The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling
Yeah I just got to that. It's amazing what graph paper can do. :-D Ok, you sound like you know this stuff, so what is the cross product used for in practical terms? The dot product I can envision being used when transforming objects easy enough, but is there some things useful for this for the direction I'm headed in (OGL programming)? Thanks for the reply! Jeremy Falcon
Jeremy Falcon wrote:
but is there some things useful for this
Many, depending on what you goals are. A) you know know the normal vector to the surface you are working with. Take that with one of the existing vector to produce a third vector and you have the local coordinate system that plane is in. B) with the normal vector of the plan you can calculate the distance an abritrary point in space is from the plane. and on and on. "Yes I know the voices are not real. But they have some pretty good ideas."
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Jeremy Falcon wrote:
what is the cross product used for in practical terms?
Consider this: you can take two vectors on a surface, that'll tell you what plane the surface resides on. Find the cross product of those, and you can decide whether or not the surface is facing the "camera"... if it's not, don't bother drawing it. Handy, eh? (you can also use it for calculating lighting / shading, or drawing nifty arrows sticking out of a surface... if you're into that sort of thing.)
---- Scripts i’ve known... CPhog 1.0.0.0 - make CP better. Forum Bookmark 0.2.5 - bookmark forum posts on Pensieve Print forum 0.1.2 - printer-friendly forums Expand all 1.0 - Expand all messages In-place Delete 1.0 - AJAX-style post delete Syntax 0.1 - Syntax highlighting for code blocks in the forums
Shog9 wrote:
you can take two vectors on a surface, that'll tell you what plane the surface resides on. Find the cross product of those, and you can decide whether or not the surface is facing the "camera"... if it's not, don't bother drawing it. Handy, eh?
How is that any different than a normal though? (And if this a stupid question, well oops :-O). Nevermind I just got it. The facing the camera part was what I missed. Thanks man. Jeremy Falcon
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Jeremy Falcon wrote:
but is there some things useful for this
Many, depending on what you goals are. A) you know know the normal vector to the surface you are working with. Take that with one of the existing vector to produce a third vector and you have the local coordinate system that plane is in. B) with the normal vector of the plan you can calculate the distance an abritrary point in space is from the plane. and on and on. "Yes I know the voices are not real. But they have some pretty good ideas."
Michael A. Barnhart wrote:
you know know the normal vector to the surface you are working with. Take that with one of the existing vector to produce a third vector and you have the local coordinate system that plane is in.
Hot damn! I'm not finished with this book yet (page 77 of 449), so I'm mainly curious. But that is extremely useful! Thanks! Jeremy Falcon
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http://mathworld.wolfram.com/CrossProduct.html[^] Todd Smith
I found that site a while back on a Google frenzy. The concept is nice but their "answers" are no less confusing than hieroglyphics. The only people that understand what they say are the people that don't need to ask the question in the first place. Thanks for the reply. Jeremy Falcon
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Jeremy Falcon wrote:
Now I wish I had graph paper.
Huh. All this technology, and I've never seen virtual graph paper. An ideal, simple, yet powerful application. Extensible. Hmmm.... Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
Visio.
"You have an arrow in your butt!" - Fiona:cool:
Welcome to CP in your language. Post the unicode version in My CP Blog [ ^ ] now.People who don't understand how awesome Firefox is have never used CPhog[^]CPhog. The act of using CPhog (Firefox)[^] alone doesn't make Firefox cool. It opens your eyes to the possibilities and then you start looking for other things like CPhog (Firefox)[^] and your eyes are suddenly open to all sorts of useful things all through Firefox. - (Self Quote)
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I found that site a while back on a Google frenzy. The concept is nice but their "answers" are no less confusing than hieroglyphics. The only people that understand what they say are the people that don't need to ask the question in the first place. Thanks for the reply. Jeremy Falcon
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Jeremy Falcon wrote:
The only people that understand what they say are the people that don't need to ask the question in the first place.
I know how that goes... :sigh:
J. Dunlap wrote:
I know how that goes...
It's sad really. Some people like to get off by acting smarter than they really are by trying to sound difficult to understand. I've seen that a LOT by some people with studying 3D math. It drives me crazy(er). Of course, I don't think that site is really geared towards those who are learning. It's really more of a reference for those who have IMO. Jeremy Falcon
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Jeremy Falcon wrote:
Now I wish I had graph paper.
Huh. All this technology, and I've never seen virtual graph paper. An ideal, simple, yet powerful application. Extensible. Hmmm.... Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
Hey, a project that would be cool to use the VCF for! :) ¡El diablo está en mis pantalones! ¡Mire, mire! Real Mentats use only 100% pure, unfooled around with Sapho Juice(tm)! SELECT * FROM User WHERE Clue > 0 0 rows returned Save an Orange - Use the VCF!
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Jeremy Falcon wrote:
Now I wish I had graph paper.
Huh. All this technology, and I've never seen virtual graph paper. An ideal, simple, yet powerful application. Extensible. Hmmm.... Marc Pensieve Some people believe what the bible says. Literally. At least [with Wikipedia] you have the chance to correct the wiki -- Jörgen Sigvardsson
Marc Clifton wrote:
All this technology, and I've never seen virtual graph paper.
AutoCAD?
Marc Clifton wrote:
An ideal
Yes
Marc Clifton wrote:
simple
No
Marc Clifton wrote:
powerful
Certainly
Marc Clifton wrote:
Extensible
Yes (but if only you could use C#...)
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Andy Brummer wrote:
The traditional cross product produces the vector perpendicular to the plane through the two vectors that you are multipling
Yeah I just got to that. It's amazing what graph paper can do. :-D Ok, you sound like you know this stuff, so what is the cross product used for in practical terms? The dot product I can envision being used when transforming objects easy enough, but is there some things useful for this for the direction I'm headed in (OGL programming)? Thanks for the reply! Jeremy Falcon
Finding the normal direction for a plane is the most common use. In addition to what Josh and others have said, not only can you use it for culling, but you can use it for lighting calculations and reflection angle. Also it is used for Snell's law for translucent objects. In basic physics it is used for all sorts of electromagneic, fluid flow and other calculations of that sort. It can also be used to find vectors in the same plane as your original vectors using (a x b) * x = 0 since this implies x = x0 a + x1 b where x is the vector you are checking. It can be really useful in 3 dimensions, but doesn't extend to higher dimensions and is harder to remember the formula for. It's too bad there isn't more info on bivectors out there.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon