Hey /g/ does anyone know why the cross product of 2 vectors is equal to the area of the parallelogram created by it. Also does anyone know the applications this could be used for?
This is /g/, not /sci/, we're here to discuss waifus, not maths
Look at the formula of the area of a parallelogram and then look at the formula for multiplying two vectors and draw conclusions from that. Otherwise, google your answer.
>>53278752
that's not even correct OP ffs...
>>53278752
vectors are spatial operators
each vector can represent a dimension or a combination of dimensions
these dimensions can be physical, but they don't have to be
the cross product of two vectors looks at how one dimension relates to one or more dimensions
the cross product of two vectors may produce a third vector
the length of this third vector:
"the length of the cross product of two vectors is equal to the area of the parallelogram determined by the two vectors"
see diagram for the geometric rationale, which isn't quite what the cross product is
conceptually, you are taking one "direction through space" and projecting it through another "direction through space", and the area through which that projection occurs is equal in "value" to the "length" of a third vector that is produced as a result of the cross product
the cross product is useful for deriving actual physical spatial products of motion through space
if the dimensions are abstract (sound volume amplitude versus treble frequency versus time) the cross product can derive formulaic combinations for more than one dimension
Are you retarded?
>>53280826
Now use Triple product to calculate the Volume!1
a=2
2
4
b=9
3
8
c=1
3
>>53281442
trying to conceptualize the triple product gives me a headache
>>53281634
there are people on /g/ that can't solve simple math
You learn that in school!
>>53280826
Do my homework too faggot, lol thanks
>>53282991
you must be over the age of 18 to use this board!
>>53280826
>vectors are spatial operators
vectors arent operators
>>53281634
3 vectors
dot product of one vector and the cross product of the other two
what is there to not understand
>>53282949
real linear algebra has one of the highest failure rates of any course in engineering school; the situation is so bad they have symposia about the problem every 3 or 4 years
>>53283086
yeah, they are
(see how easy that was?)
>>53283734
clear as mud
you'll be a great middle school teacher someday
>>53278752
||AxB|| = ||A||||B||sin(theta)
Pretty straight forward. The cross product is simply the multiplication of two vectors, but since the height of the object is influenced by not just the magnitude of the vectors, but the angle between them, the cross product reflects that.
>>53278752
I would use matlab
>>53284704
>cross product is simply the multiplication of two vectors
>simply
how do you "multiply" a direction
good luck answering this you glib regurgitator
>>53278752
> why the cross product of 2 vectors is equal to the area of the parallelogram created by it
Well, the area A = height * base
Now to get height of a specefic side, project it into the base. So height = side * sinX [X = the angle between side and base]
so, A
=(side*sinX) * base
=a sinX * b
=ab sinX
Now the unit is changed because you multiplied two lengths. If the units are 1i an 1j their product is
i*j
=1*1*sin90 (i*j)
=1k
[The k unit is a unit vector that creates 90 degrees with both i and j. i and j are both unit vectors along the X and the Y axis respectively. The area that's created by the cross product has a unit of k, perpendicular to the XY plane.]
So by definition both multiplication are the same. The cross product is introduced to determine properties such as this (i.e. Area of a parallelogram). Cross product is useful only when the direction of the product is not the same i.e. Torque. Dot product is used only where the direction of the two vectors are not changed, i.e. Angular momentum.
It's hard to define a definition, you will get used to it soon.
> used for
This is a very fundamental concept. You will need it for physics mainly. So giving examples is pointless.
>Lastly
Install Gentoo.
>>53285086
Vector
>>53278752
go ask /a/
>>53285307
>Dot product is used only where the direction of the two vectors are not changed
*direction of the two quantity. I meant the products of dot multiplication is NOT a vector if you are unclear
>math
This is a technology board. Eat shit faggot
>>53285486
It is tech relevant. More relevant than Speccy threads
>>53284574
>you'll be a great middle school teacher
No, I am not majoring in education.
I still don't see what you're not understanding though. It's not an advanced concept of linear algebra.
>>53285086
Same way you multiply the sides of a room to get a square footing.
>>53285486
Still more technology than animu wallpaper threads.
>>53284574
>>>53282949
>real linear algebra has one of the highest failure rates of any course in engineering school; the situation is so bad they have symposia about the problem every 3 or 4 years
Lmao linear algebra is easy as shit. I'm at a top tier engineering school and nobody has problems with Lin alg
>>53284574
what's the "right" way to teach LinAlg currently believed to be?
my undergrad class used the Strang book, which dealt with things mostly at an abstract level, and basically pushed the idea of thinking about the relations between the subspaces, etc.
the course basically built up towards the what, how, and why of SVD and then into applications of it and shit like FFT, etc.
pic related haunted my dreams for that trimester.
>>53285486
The way cross products are calculated just happens to be the same way the area of the parallagram would be calculated. Its kind of like how integral of a curve coincides with the area under a curve. Its an easy way to visualize a complex concept but can make you lose objectivity and hurt you in the long run.
>>53278752
>Hey /g/ does anyone know why the cross product of 2 vectors is equal to the area of the parallelogram created by it.
Yes. Because it's defined that way
>>53278752
MARTIN is this you?
>>53286246
"Report of the Technology Subgroup"
http://matrix.skku.ac.kr/ilas/er/data/rts.htm
Warning: Site contains actual technical materials unrelated to weaboo shit.
>>53287285
wow, that is fucking awful.
"tech" focus in LinAlg is like trying to grind the mechanics of arithmetic into kids without trying to teach them algebra.
In the real world, you have cheap hardware that can do ridiculous numbers of matrix operations every second, and the only real challenge is understanding in the abstract what transformations (and why) need to be applied to data.
>>53287451
they're suggesting using technology as a tool towards comprehension; not as a replacement for instruction or comprehension
"only real challenge"
you've also misunderstood the nature of linear algebra as a developing technique; the rote computations done by computerization only scratches the surface of methods being added to linear algebra
