Simple Vector Question

1. Aug 1, 2004

Physics_Student

I'm having trouble with what should be a simple question!

Let a = (2,4,-2) and b = (4,-2,2)

I need to be able to express a as the sum of two vectors, one parallel to b and the other perpendicular to b.

Thing is, I haven't the foggiest idea where to start! Any ideas?

Thanks

2. Aug 1, 2004

Muzza

The vector parallel to b is called the "projection of a onto b". There is a formula for it, and it should be covered in any basic book on linear algebra (at least in the cases of the vectors being in R^2 or R^3). proj(a, b) = (a.b)/(b.b) * b, (but obviously it's no good to just know the formula, so get yourself a book) ;).

3. Aug 1, 2004

TenaliRaman

erm,
u mean the component of a parallel to b is "the projection of a onto b" right?

anyways as muzza said the parallel component of a comes as a projection of a onto b and the entire thing can be written as,
a = [(a.b)/b^2] b + (a - [(a.b)/b^2] b)

the first component is parallel to b and the second component is perpendicular to b.

-- AI

4. Aug 1, 2004

Muzza

Yes, I figured that was understood.

5. Aug 1, 2004

Physics_Student

Thanks for the speedy replies.

I get where the parallel component comes from, but I don't understand where the perpendicular component comes from?

6. Aug 1, 2004

Muzza

Let a_p be the aforementioned vector parallel to b, and a_o be the perpendicular vector. Then a = a_p + a_o <=> a_o = a - a_p = a - (a.b)/(b.b) * b.

7. Aug 1, 2004

Physics_Student

Thanks, that explained it very clearly. Can't believe I didn't notice it was that simple.

Thanks

8. Jan 7, 2012

aswinsp

there should be some another method

Last edited: Jan 7, 2012
9. Jan 7, 2012