Calculating Orthogonality of Binormal Vector with Dot Product

[ronaldor9]

How can I show that the binormal vector is orthogonal to the tangent and normal vector. I know i should use the dot product to determine this, however i do i actually go about doing it?

 

[MathematicalPhysicist]

Well, if T is the tangent vector, N the normal to T, and B=TxN (cross product of the two), then you should use the fact that a*(bxc)=det[a,b,c] where * is the dot product.
I.e it equals the determinant of the vectors (in rows) a,b and c.

 

[Hurkyl]

How can I show that the binormal vector is orthogonal to the tangent and normal vector.

Isn't that (most of) the definition of the binormal vector?


Maybe you're using a different exposition than I would expect -- what definition of "binormal vector" is your class using?

 

[ronaldor9]

this was a problem in a my book. MathematicalPhysicist, that is what i want to do however is there a way to show it is perpendicular without any specific values for the components of the vector?

 

[mathman]

If you follow the reasoning of Mathematical Physicist, when you take the dot product of either the normal or tangent vector with the binormal, the determinant in question has two identical rows, so that its value is 0.