Expressing an Arbitrary Vector in Terms of Noncoplanar Vectors

[EngageEngage]

Homework Statement

Show that an arbitrary vector V can be expressed in terms of any three noncoplanar vectors, A, B, C, according to:

V = [V,B,C]A/[A,B,C] + [V,C,A]B/[A,B,C] + [V,A,B]C/[A,B,C]

Homework Equations

A Hint is given:
We know that V can be expressed as aA + bB +cC; to find a, take the scalar product of V with BxC

The Attempt at a Solution

I tried to solve this one by relating the projections of V to 3 arbitrary vectors, A, B, C, but I couldn't get to the answer above. I'm also not sure how the hint will help me either. Could someone please help me get started on this because I am all out of ideas.

thanks

 

[Dick]

I'm assuming [V,B,C] is the triple product V.(BxC), right? Then what is [V,B,C]? It's (aA+bB+cC).(BxC). BxC is perpendicular to B and C, so those parts of the dot product are zero. This leaves you with [V,B,C]=aA.(BxC)=a*[A,B,C]. Put that into your formula and treat the other two terms the same way.

 

[EngageEngage]

Wow, so the hint gave it away -- i can't believe i didn't see that. Thanks a lot for the help though!