Vector Cross Product: Understanding the Perpendicular Result

[parshyaa]

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

 

[fresh_42]

May I ask you, what kind of answer do you expect? The way you phrased the question is a tempting pass-play to create jokes.

 

[mfb]

It is a useful operation that way.

 

[Stephen Tashi]

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

The cross product is useful in physics to solve problems involving torques. https://en.wikipedia.org/wiki/Torque

If you turn a bolt with a wrench whose handle is (perfectly) perpendicular to the shaft of the bolt and exert a force that is (perfectly) normal to the plane of the handle and the shaft then you can calculate the torque on the bolt without worrying about vectors. However, in real life situations, the handle of a wrench isn't perfectly perpendicular to the bolt and the force exerted on the handle isn't perfectly normal to the plane of the shaft and the handle.

The cross product between the vector L describing the handle and the vector F representing the force gives a vector $\tau$ representing the torque on the shaft of an "imagined" bolt. If you project the vector $\tau$ on a vector S that represents the shaft of the real bolt, you get the torque on the real bolt.

If the bolt is "right hand" threaded, the projection of $\tau$ on S is a vector that actually points in the direction that the applied torque tends to screw (or unscrew) the bolt into something.

 

[Dale]

Several off topic posts have been removed. Please try to keep responses on topic

 

[parshyaa]

• I just want to know how it came to their mind, there is a idea for every thing , just like we know that A + B = C (A,B,C are vectors) and this can be represented by the diagram given below:-

• If someone ask, why you represented addition of vector like given in above diagram.
• Then there is a answer , yes we know this , take a football or box or an object, if you apply a force (which is a vector) on it from left side it will move right side and if you do it from right side it will move towards left side and if you hit it simultaneously it will go between them, this idea exactly represents the above diagram.

• therefore I think that there may be a good reason for the cross product question

 

[robphy]

The [wedge] cross product of vectors form a bivector.
In three dimensions, this bivector has three components and can be though of as a https://en.wikipedia.org/wiki/Pseudovector . (This won't work in higher dimensions.)

Hmmm... The following is probably incomplete... but I think it'll lead to the full story...
If you wedge product the bivector result with any vector that is coplanar with that bivector, you get zero.
One probably needs to invoke linearity, antisymmetry, and associativity [oops... it's non-associative (..Jacobi identity)] to finish it off...

 

[fresh_42]

I just want to know how it came to their mind, there is a idea for every thing , just like we know that A + B = C (A,B,C are vectors) and this can be represented by the diagram given below

I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: $\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.$

If someone ask, why you represented addition of vector like given in above diagram.

That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.

Then there is a answer ... therefore I think that there may be a good reason for the cross product question

No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.

 

[parshyaa]

For

I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: $\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.$That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.

For me its a cool invention , but I got your point.

 

[fresh_42]

For me its a cool invention , but I got your point.

A difference on the philosophical point of view. Sometimes as well the difference between more "applied" and more "pure" scientists.
Your question brought me to think about the following: Who first used it and in which context? (That's why I was asking in #2... and to keep me from joking like saying: It has been to annoy physicists )

Has it been an ancient geometer like Euklid or Aristoteles? Was it Graßmann? Descartes? To be honest, I don't know and have not really an idea where to search for the answer.

Edit: The book about history I have (J. Dieudonné) says: Graßmann and he was indeed driven by geometric considerations like generalizing the one-dimensional length of a vector.

 

[Dale]

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

I think of it this way. We have a different product, the dot product, which takes a pair of vectors and gives a scalar. So we want the cross product to give a vector.

Two vectors define a plane of vectors which are linear combinations of the two vectors, so there is no point in making our cross product give any of those vectors. So if you want to get out of that plane, then the only sensible vector is a vector normal to the plane (any other vector is a linear combination of that and a vector in the plane). That vector is necessarily perpendicular to any vector in the plane, including the original two.

 

[chiro]

Hey parshyaa.

There are many developments for the cross product but one of the main ideas was to apply multiplication and division to vectors and not just numbers.

The story is deep but a teacher by the name of Hermann Grassman (a german from the 1800's) looked at solving this problem of multiplying and dividing vectors so that you could do a multiplication and then a division (like A*B/B = A) on a vector and developed a lot of the idea for geometric algebra (i.e. applying arithmetic algebra to geometric objects).

It has been extended significantly since then by many others and it has a lot of physical intuition (as has been mentioned above) but the idea of being able to multiply and divide vectors so that it works like a normal number division has a lot to do with the thinking behind it and its generalization in multiple dimensions.