Cross product of difference and sum of two vectors

In summary, the conversation is about proving the statement (a-b)x(a+b)=2a x b and its geometric interpretation. The solution involves using the fact that a+b and a-b are the diagonals of a parallelogram with a and b as sides. The final answer is 2(a x b), which is equal to (2a) x b or 2a x b. The conversation also includes a clarification about the linear operator property of the cross product.
  • #1
msslowlearner
33
0

Homework Statement


show that:
( a - b ) x (a + b ) = 2a x b

and wat is its geometric interpretation ??
I'm not sure what's wrong, but i somehow got the value as 2a and not wat was required... PLease help.




Homework Equations


Since this is a proof, the answer I've arrived at is wrong. How do I arrive at the solution ?


The Attempt at a Solution


First of all, i took the cross product and tried proving the statement, but couls arrive at it whatsoever. I ended up with a 2a actually.
As for the second part, with the parallelogram idea, i really could not figure out how to find 2a x b. After analysis, I arrived at 2a , but am stuck there . please help.
 
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  • #2
Can you show your work? I get the answer required in two lines of algebra.
 
  • #3
Geometrically, a+ b and a- b are the diagonals of a parallelogram having a and b as sides. The geometric content of your equation is "If a and b are sides of a parallelgram, P, then the parallelogram having its diagonals as sides has area twice the area of P".
 
  • #4
thanks hallsofivy..., now it appears more clear ... i messed up my vector diagrams so badly i didn't see the actual thing .. for the first part, i got 2 (a x b). is it equal to
2a x b ??
 
  • #5
msslowlearner said:
thanks hallsofivy..., now it appears more clear ... i messed up my vector diagrams so badly i didn't see the actual thing .. for the first part, i got 2 (a x b). is it equal to
2a x b ??

No, 2(a x b) is not same as 2a x b. :smile:

And the question you have posted should be:-
show that : (a-b)x(a+b)=2(a x b)
 
  • #6
should be then .. but the textbook says 2a x b. or maybe i read it wrong !
 
  • #7
Sorry Pranav-Arora, but 2(a x b) = (2a) x b = 2a x b.
The cross product is a linear operator.

Note that a x a = 0 and that a x b = - b x a.
 
  • #8
I like Serena said:
Sorry Pranav-Arora, but 2(a x b) = (2a) x b = 2a x b.
The cross product is a linear operator.

Note that a x a = 0 and that a x b = - b x a.

Sorry, You're right. :smile:
 
  • #9
thanks people :)
 

What is the cross product of difference and sum of two vectors?

The cross product of difference and sum of two vectors is a mathematical operation that results in a new vector that is perpendicular to both of the original vectors. It is a vector quantity and is denoted by the symbol "×".

How is the cross product of difference and sum of two vectors calculated?

The cross product is calculated by taking the determinant of a 3x3 matrix composed of the unit vectors i, j, and k and the components of the two vectors. The resulting vector is the cross product of the two original vectors.

What is the physical significance of the cross product of difference and sum of two vectors?

The cross product has many physical applications, such as determining torque and angular momentum in physics, and calculating the direction of the magnetic field in electromagnetism. It also has applications in computer graphics and engineering.

What are the properties of the cross product of difference and sum of two vectors?

The cross product has several important properties, including being anti-commutative (meaning the order of the vectors matters), distributive, and following the right-hand rule. It also has a magnitude equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them.

How is the cross product of difference and sum of two vectors related to the dot product?

The cross product and dot product are two different operations involving vectors. The dot product results in a scalar quantity, while the cross product results in a vector quantity. However, they are related through the distributive property, and the dot product of two perpendicular vectors is equal to zero, while the cross product results in a vector perpendicular to both original vectors.

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