Question about property of Cross Product

In summary, the cross product of two vectors in R3 is a vector that is orthogonal to both vectors and is perpendicular to the plane spanned by the two vectors. This plane is not bounded and can be visualized as a parallelogram created by the two vectors. The order in which the elements of the vectors are multiplied to get the cross product may seem strange, but it is necessary for the resulting vector to be perpendicular to the original vectors. The resulting vector is only from the point where the two original vectors intersect.
  • #1
zeion
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1

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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  • #2
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.
 
  • #3
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?
 
  • #4
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
 

FAQ: Question about property of Cross Product

1. What is the definition of cross product?

The cross product, also known as the vector product, is a mathematical operation that produces a vector perpendicular to both of the vectors being multiplied. It is denoted by the symbol "x" and is defined as A x B = ||A|| ||B|| sinθ n, where A and B are the two vectors, θ is the angle between them, and n is the unit vector perpendicular to both A and B.

2. How is the cross product different from the dot product?

The dot product results in a scalar quantity, while the cross product results in a vector quantity. The dot product is also commutative, meaning A · B = B · A, whereas the cross product is anti-commutative, meaning A x B = -B x A. Additionally, the dot product measures the projection of one vector onto another, while the cross product measures the perpendicular component of one vector to another.

3. What are some real-life applications of the cross product?

The cross product has numerous applications in physics, engineering, and computer graphics. Some examples include calculating torque in mechanics, determining the direction of magnetic fields, and creating 3D animations in computer graphics.

4. How do you calculate the magnitude of the cross product?

The magnitude of the cross product can be calculated using the formula ||A x B|| = ||A|| ||B|| sinθ, where ||A|| and ||B|| are the magnitudes of the two vectors and θ is the angle between them.

5. Can the cross product be performed on vectors in any dimension?

No, the cross product is only defined for vectors in 3D space. It is not possible to calculate the cross product of vectors in higher or lower dimensions.

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