Question about property of Cross Product

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Homework Help Overview

The discussion revolves around the properties of the cross product of vectors in three-dimensional space, specifically focusing on the geometric interpretation of the cross product and the plane spanned by two vectors.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand why the cross product of two vectors is orthogonal to the plane they span, expressing confusion about the geometric representation of this plane. Participants explore the relationship between the vectors and the parallelogram formed by them.

Discussion Status

Some participants have provided clarifications regarding the geometric interpretation of the vectors and the parallelogram, while others continue to question the implications of the cross product's definition and the order of multiplication. Multiple interpretations of the concepts are being explored.

Contextual Notes

The original poster expresses a lack of familiarity with the subject, indicating that they may have misunderstandings about the properties of the cross product and the geometric implications of vector operations.

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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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