# What does cross product of vectors actually mean?

I understand that dot product of vectors means projecting one vector on to the other. But I don't understand what is the physical significance of a cross product? I have read that cross product gives the area of the parallelogram which has each of the vectors as its sides....but why do we want to find the area of the parallelogram?

Related Other Physics Topics News on Phys.org
I understand that dot product of vectors means projecting one vector on to the other. But I don't understand what is the physical significance of a cross product? I have read that cross product gives the area of the parallelogram which has each of the vectors as its sides....but why do we want to find the area of the parallelogram?
It is a geometric construction that provides a vector perpendicular to two other vectors and that vector has length the area of the parallelogram. If you understand that, then you understand everything there is to know about cross products.

I understand that you're looking for some kind of geometric meaning, but there isn't really one. It is just a construction that comes in handy in applications. So what you really want to know is how the cross product is used in applications.

In my opinion, there is no natural significance of the cross product that is easy to understand. It is however very useful, but then you should check the specific uses.

1 person
If you don't want to find the area of a parallelogram, then don't use the cross product...

Math is math, it need not correspond to any physical situation. The math we study often does correspond to physical situations because that is the math we find useful. The physical significance of a mathematical operation is dependent on the physical situation we are modeling or describing with that math.

Cross products are a kind of measure of "difference" between two vectors (in opposition to the dot product which is a measure of the "sameness" between two vectors). With a cross product the more perpendicular your two vectors are the higher your cross product's magnitude will be. If your two vectors are parallel this measure of "difference" will be zero. If they are completely perpendicular then this measure of "difference" will be the maximum, the product of the two vector's magnitude. It will also point in the direction that is furthest (in a sense) from the two vectors.

One physical situation that is described by the cross product is when you have a torque. Your force vector is applied and there is lever vector to the rotation axis. The torque is the cross product of these two vectors. If your force is parallel with your lever, there will be no torque. If your force is perpendicular to the lever then your torque will be at a maximum. In between these two extremes the torque can be many different values. The torque is a quantity that is the "difference" between the lever arm vector and force vector. It is also the value of the area of a parallelogram that is formed by the two vectors of the force and lever.

1 person
robphy
Homework Helper
Gold Member
The magnitude of the cross product of two vectors could be interpreted as a measure of the "linear independence" of the two vectors.

In many physical situations, the cross product is an indication that...
[abstractly] you are dealing with a more complicated "directed quantity" (www.google.com/search?q=cross+product+differential+form) than merely a typical [polar] vector.

http://www.math.oregonstate.edu/bridge/mathml/dot+cross.xhtml (with applets)
http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf

http://math.stackexchange.com/questions/61526/help-understanding-cross-product

1 person
256bits
Gold Member