Question about property of Cross Product

AI Thread Summary
The discussion centers on understanding the properties of the cross product of two vectors in R3, specifically how the result is orthogonal to both vectors and the plane they span. The user initially confuses the orientation of the cross product, thinking it might be parallel to the plane formed by the two vectors, but later clarifies that it is indeed perpendicular. The concept of the parallelogram formed by the two vectors is highlighted, emphasizing that the vectors serve as two sides of the parallelogram while the other two sides are parallel to them. Questions arise about the geometric representation of the plane and the order of multiplication in the cross product calculation. Visual aids, such as those found on Wikipedia, are suggested for better understanding.
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Homework Statement




Hello. I am relatively new to this subject please forgive my incompetence.
Please correct me if I have misunderstandings.

I understand that the cross product of two vectors (say A and B) in R3 is a vector that is orthogonal to both A and B. But how A x B be orthogonal to the plane spanned by A and B?

I don't understand what this plane spanned by A and B should look like geometrically.

Wouldn't A x B be parallel to the plane spanned by A and B?
 
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Think of your two vectors A and B positioned with their tails together and assume they are not parallel. Think of the parallelogram created using those vectors as two sides of the parallelogram. That parallelogram is part of the plane spanned by A and B. A x B is perpendicular to that plane.
 
Oh okay I got that mixed up then. Thanks.

So then the plane by spanning A and B has the vectors A and B as the boundaries right?
Why does it make a parallelogram when there is only two vectors?

Now my next question is how to understand the weird order in which you multiply the elements of A and B to get A x B?

The vector gotten from A x B is only from the point where A and B intersect?
 
A plane has no boundaries. The vectors are two sides of the parallelogram, and the other two sides can be drawn parallel to the original two vectors to complete the parallelogram.

Wikipedia has some nice pictures which may help clarify things:
http://en.wikipedia.org/wiki/Cross_product
 

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