# Explain the cross product.

1. Aug 13, 2013

### global

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.

2. Aug 13, 2013

### SteamKing

Staff Emeritus
3. Aug 13, 2013

### global

I don't see how this helps.

4. Aug 13, 2013

### Staff: Mentor

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.

5. Aug 13, 2013

### global

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. Aug 13, 2013

### micromass

Staff Emeritus
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.

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. Aug 13, 2013

### global

Now that I got the first question. Can you guys answer question 2 whys is i x j =k and not -k?

8. Aug 13, 2013

### robphy

Possibly useful:
http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf

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. Aug 13, 2013

### global

I am sorry but can you please explain, I already saw the wikipedia article and didn't understand it.

10. Aug 13, 2013

### mathman

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. Aug 13, 2013

### global

i don't how get how this works though by convention. Why does i * j=k why can't it equal -k.

12. Aug 13, 2013

### robphy

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. Aug 13, 2013

### global

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. Aug 13, 2013

### lurflurf

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. Aug 13, 2013

### global

I have no idea what you just said.

16. Aug 13, 2013

### Mandelbroth

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. Aug 13, 2013

### global

I have no idea what you are doing.

18. Aug 13, 2013

### Mandelbroth

What don't you understand?

19. Aug 13, 2013

### robphy

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. Aug 14, 2013

### mathman

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.