Cross product of difference and sum of two vectors

[msslowlearner]

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.

 

[Pi-Bond]

Can you show your work? I get the answer required in two lines of algebra.

 

[HallsofIvy]

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".

 

[msslowlearner]

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 ??

 

[Saitama]

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.

And the question you have posted should be:-
show that : (a-b)x(a+b)=2(a x b)

 

[msslowlearner]

should be then .. but the textbook says 2a x b. or maybe i read it wrong !

 

[I like Serena]

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.

 

[Saitama]

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.

 

[msslowlearner]

thanks people :)