Explain the cross product.

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Why does the cross product produce a vector and why is that vector perpendicular to the other vectors?
I understand how to calculate a cross product, but why for instance is the cross products of two vectors another vector that is perpendicular to it. Can you prove or explain this to me in anyway. There are two parts I want you to answer here:
1. How this calculation yields another vector
2. Why this vector is perpendicular to the other two vectors.
After i have understood the 2-part question above. Why is the cross product of the unit vectors i x j=k and j x i= -k? After all the answer of a cross product is vector that is perpendicular to the other vectors. In the example i x j = k, the vector -k is just as perpendicular to these two vectors as the vector k, why can't the answer of i x j be equal to also -k. In the same way why is j x i= -k, the vector k is just as perpendicular as -k, so why can't j x i be equal to vector k. If this is just convention that scientist agree upon, then i don't understand how we can expect nature to "conform" to standard conventions that everybody has agreed upon.
 

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  • #3
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I don't see how this helps.
 
  • #4
jtbell
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i don't understand how we can expect nature to "conform" to standard conventions that everybody has agreed upon.
You've got it backwards... we design the conventions so as to "conform" to nature, that is, to give a consistent mathematical description that agrees with what we observe from nature. It turns out the cross product is useful in a number of contexts, most importantly rotational kinematics and dynamics, and magnetic forces and fields.
 
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  • #5
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SO the only reason why the cross product yields a vector and why that vector is perpendicular is just an attempt to describe what is already in nature? Because I have always thought of it like this. One day a bunch of mathmaticans just said randomly, hey let make a new operation of two vectors called the cross product and define it so that the answer is another perpendicular vector, for no particular reason at all. How what you are saying is, that a bunch of mathmaticians observe a bunch of related phenomenon, and wrote the definition to describe these events.
 
  • #6
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Because I have always thought of it like this. One day a bunch of mathmaticans just said randomly, hey let make a new operation of two vectors called the cross product and define it so that the answer is another perpendicular vector, for no particular reason at all.
This is the kind of misconception that many students have. But it is so far from the truth. Almost everything we do in mathematics is in some way rooted in trying to understand nature.

How what you are saying is, that a bunch of mathmaticians observe a bunch of related phenomenon, and wrote the definition to describe these events.
That is exactly what mathematics is. We observe phenomena which look very similar, and we abstract it and make it rigorous. Nothing really comes out of thin air, it always has some kind of underlying reason, even though that is not always very clear to everybody.
 
  • #7
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Now that I got the first question. Can you guys answer question 2 whys is i x j =k and not -k?
 
  • #8
robphy
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Possibly useful:
http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf

The Wikipedia link is also useful... but requires some careful reading.

One point needs to be made... The feature of the cross product being a vector perpendicular to the factors depends on its use for 3d-vectors. The [oriented] parallelogram formed is the more important feature.
 
  • #9
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Possibly useful:
http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf

The Wikipedia link is also useful... but requires some careful reading.

One point needs to be made... The feature of the cross product being a vector perpendicular to the factors depends on its use for 3d-vectors. The [oriented] parallelogram formed is the more important feature.
I am sorry but can you please explain, I already saw the wikipedia article and didn't understand it.
 
  • #10
mathman
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Now that I got the first question. Can you guys answer question 2 whys is i x j =k and not -k?
This is basically by convention. ixj = k, jxk = i, kxi=j. If you reversed one sign, you would need to reverse the others. Essentially you need this so that a set of mutually perpendicular vectors, defined with cross product, would not change when rigidly rotated.
 
  • #11
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i don't how get how this works though by convention. Why does i * j=k why can't it equal -k.
 
  • #12
robphy
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the choice of which direction perpendicular to the plane the unit-vector "k" should point
is associated with the right-hand-rule (convention)
...so, diagrams should also be drawn with a "right-handed-coordinate system"
 
  • #13
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is the right hand rule have something to do withnature. for example in nature does a vector pointing to the right (i direction) interacting with a vector going up (j direction) always produce a vector coming in at you( k direction). If nature had a vector pointing to the right (i direction) interacting with a vector going up (j direction)to produce a vector coming out at you(negative k direction). would i*j=-k instead of k
 
  • #14
lurflurf
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The cross product is the only possible (up to multiplications by a constant) product. It makes sense that it is something we would care about. Likewise is we show that the perpendicular to a pair of vectors is a product it must be the cross product.
 
  • #15
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The cross product is the only possible (up to multiplications by a constant) product. It makes sense that it is something we would care about. Likewise is we show that the perpendicular to a pair of vectors is a product it must be the cross product.
I have no idea what you just said.
 
  • #16
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Now that I got the first question. Can you guys answer question 2 whys is i x j =k and not -k?
I believe it comes from a previous idea called the "quaternion." The basic idea is to have ##i^2=j^2=k^2=ijk=-1##, and then use the 3 separate "imaginary" units as unit vectors. If we right multiply ##ijk=-1## by ##k##, we see that ##-ij=-k##, and therefore ##ij=k##.

For our purposes, the right hand rule is natural simply because of how we orient axes.
 
  • #17
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I have no idea what you are doing.
 
  • #18
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I have no idea what you are doing.
What don't you understand?
 
  • #19
robphy
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is the right hand rule have something to do withnature. for example in nature does a vector pointing to the right (i direction) interacting with a vector going up (j direction) always produce a vector coming in at you( k direction). If nature had a vector pointing to the right (i direction) interacting with a vector going up (j direction)to produce a vector coming out at you(negative k direction). would i*j=-k instead of k

The right-hand rule is a convention. (We could have chosen the left-hand rule, and appropriately changed all of the relevant signs.... but someone chose the right-hand.)
Similarly, someone [effectively] declared the sign of the electron to be negative.

The point is...
if we stare a parallelogram (say your computer monitor),
we want to distinguish
a clockwise orientation [vertical-up, then horizontal-right] from
a counterclockwise orientation [horizontal-right, then vertical-up].
Note that the order is important... and that swapping the order introduces a relative minus sign.
That's why the cross-product [an antisymmetric operation on an ordered pair of vectors] is an appropriate mathematical object for this situation.

Someone chose a convention that counterclockwise is positive [like in the polar coordinates]
and that with the right-hand rule, your right-hand thumb will point out of the plane of the computer monitor
[...and because we live in three-dimensions, your thumb points in the unique direction (or minus that direction) that is perpendicular to the vectors in your plane... in higher-dimensions, your thumb isn't enough to point to directions perpendicular to the plane].
 
  • #20
mathman
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i don't how get how this works though by convention. Why does i * j=k why can't it equal -k.
It is by convention. If you have ixj = -k, then for consistency jxk = -i and kxi = -j. You can have either a right hand rule or a left hand rule. You just can't mix. By convention, the right hand rule is used. It is a mathematical convention, not a law of nature.
 
  • #21
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Well, there are two equivalent definitions for the cross product. One is more algebraic and deals with the coordinates:
[itex](x_1, x_2, x_3) × (y_1, y_2, y_3) = (x_2 y_3 - x_3 y_2, x_3 y_1 - x_1 y_3, x_1 y_2 - x_2 y_1)[/itex].

The other definition is more geometric. The cross product is a vector of magnitude equal to the area of the parallelogram between the two vectors, and with direction perpendicular to their common plane, pointing up according to the right-hand rule. The two definitions can be proven to be equivalent (the proof is just straightforward but tedious algebra).

The reason for defining the cross product that way is because it is useful in many contexts.

For example, in physics, many quantities are best generalized as vectors using the cross product:
Torque: [itex]\vec{τ} = \vec{r} × \vec{ F }[/itex]
The torque measures how hard it is to give a rotational acceleration to a spinning object like a door that's being pushed upon, and the above formula relates it with the force required to obtain this acceleration if one pushes at distance r from the pivot. The torque points parallel to the axis of rotation, and points up if the acceleration is counter-clockwise, and down if it's clockwise.

Magnetic force: [itex]\vec{F_m} = q\vec{v}×\vec{B}[/itex]
Here [itex]\vec{F_m}[/itex] is the force on a charge moving with velocity [itex]\vec{v}[/itex] in a magnetic field [itex]\vec{B}[/itex]. It is an experimental fact that charges moving parallel to the magnetic field are not affected ([itex]\vec{F_m} = \vec{0}[/itex]) and charges moving perpendicularly to the magnetic field are affected by the maximum force of [itex] F_m = qvB [/itex], which is perpendicular to both the velocity and the magnetic field. Both of these formulas are special cases of the more general cross product formula.

In mathematics, cross products can be used to find vectors perpendicular to a plane, which is useful for computing shortest distance and projections. They can also be used to find the volume of a 3-d parallelepiped formed by [itex]\vec{u}, \vec{v}, \vec{w}[/itex], which is [itex](\vec{u}\times \vec{v} ) \cdot \vec{w} [/itex]

I hope this is sufficient examples to show that the cross-product is not just a mathematical invention with no purpose.
 
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  • #22
As you may have learned in a physics class that involved electricity and magnetism, orientation and left handed and right handedness are vital to understand the subject. Its sort of like in a wire, you can imagine positive charges running one way, or negative charges running the opposite direction, both imply the same thing, but we must choose one in order to get a mathematically continuous system that works.
 
  • #23
chiro
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The cross product has a quite a long history.

One way to look at it is to look at multiplication of vectors. We can multiply numbers easily including integers, rationals, reals, and complex numbers. But what about numbers in higher dimensions?

Well it turns out that doing so leads to things like cross products, quaternion products, and other higher dimensional algebras.

There was a famous mathematician (who didn't get nearly as much credit as he should have gotten) called Hermann Grassman who actually invented and formalized the idea of a general vector algebra and formed the idea of the bivector using a standard geometric product.

This bi-vector had two components: one was a vector and one was a scalar.

In modern mathematics that is taught in high school and university, we call the vector component of the appropriate geometric product the cross product and the scalar component of the bi-vector the inner product.

If the cross product contains no zero-vectors in a x b = c then it can be inverted to get a in terms of b and c or a in terms of b and c just like you do when you use real or complex numbers in place of 3D vectors.

If you take the formal approach of trying to create an algebra with the right properties of multiplication, then you end doing what Grassmann and Sir William Rown Hamilton did to get the results for vector algebra and quaternion algebra.

However if you want an application to applied arenas, then you look to areas like physics that give these algebras some context in the physical realm.

If you want to understand more about the cross product and where all the geometric algebras come from, I'd suggest you get a book that talks about the results of Hermann Grassman because it will make a lot more sense to you rather than looking at a lot of the modern accounts that just state definitions.
 
  • #24
robphy
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There was a famous mathematician (who didn't get nearly as much credit as he should have gotten) called Hermann Grassman who actually invented and formalized the idea of a general vector algebra and formed the idea of the bivector using a standard geometric product.

This bi-vector had two components: one was a vector and one was a scalar.
If I am not mistaken, the bivector is just the (exterior product) vector part of the geometric product
http://en.wikipedia.org/wiki/Bivector
 
  • #25
chiro
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The true bivector should be a combination of a vector and a scalar.

In that link you posted, take a look at the various examples of the geometric product that contain both an inner and outer product and you'll see what I mean.
 

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