Expressing Vector a as the Sum of Two Vectors Parallel and Perpendicular to b

[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

 

[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) ;).

 

[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

 

[Muzza]

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

Yes, I figured that was understood.

 

[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?

 

[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.

 

[Physics_Student]

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

Thanks

 

[aswinsp]

there should be some another method

 

[aswinsp]

misread it

 

[Ryan_m_b]

aswinsp thank you for your contribution but if you look at the time stamp above people's names you will see that this thread is 8 years old. Posting in such an old thread is called necroposting and is not allowed.