Determine if three vectors form a right hand triple?

[fredrick08]

Homework Statement

ok can someone tell me how to determine if three vectors form a right hand triple? and why does (a-b)x(a+b)=2(axb)...

please someone help, I am really confused on how to do these... arent just three vectors a right hand triple if the dot product between them all is 0?? I am not sure though please someone help.. and the second one.. the must be a formula or a property i can't find... thanks

 

[HallsofIvy]

Homework Statement

ok can someone tell me how to determine if three vectors form a right hand triple? and why does (a-b)x(a+b)=2(axb)...

please someone help, I am really confused on how to do these... arent just three vectors a right hand triple if the dot product between them all is 0?? I am not sure though please someone help.. and the second one.. the must be a formula or a property i can't find... thanks

Well, first of all what is the definition of "right hand triple"? That would seem a good place to start!

For the second, I assume that you are talking about the cross product of vectors. (a- b)x(a+ b)= what if you just go ahead and multiply it out (the distributive and associative laws of multiplication are true for the cross product). Now remember that cross product if anti-commutative. What is axa? What is bxb?

 

[Architeuthis Dux]

I can provide some guidance on how to determine if three vectors form a right hand triple and explain the relationship between (a-b)x(a+b) and 2(axb).

To determine if three vectors form a right hand triple, we can use the right hand rule. This rule states that if you curl the fingers of your right hand in the direction of the first vector, then the second vector, the resulting thumb will point in the direction of the cross product of the two vectors. If the third vector is in the same direction as the resulting thumb, then the three vectors form a right hand triple.

As for the relationship between (a-b)x(a+b) and 2(axb), we can use the properties of the cross product. The cross product of two vectors, a and b, is perpendicular to both a and b and has a magnitude of |a||b|sinθ, where θ is the angle between the two vectors. Therefore, (a-b)x(a+b) has a magnitude of |a-b||a+b|sinθ, while 2(axb) has a magnitude of 2|axb|. Since |a-b| = |a+b|, we can see that (a-b)x(a+b) and 2(axb) have the same magnitude. Additionally, the direction of (a-b)x(a+b) is opposite to that of 2(axb), which is why we use the negative sign in the formula (a-b)x(a+b) = -2(axb). This relationship is based on the properties of the cross product and can be used to simplify calculations involving cross products.

In conclusion, it is important to understand the right hand rule and the properties of the cross product in order to determine if three vectors form a right hand triple and to understand the relationship between (a-b)x(a+b) and 2(axb). I hope this helps clarify your confusion.