Finding Values of r and q to Satisfy H

[mccoy1]

Homework Statement

H is subset of R^3 dfined by H = {(a,b,c) element of R^3: a+2b-c = 0}. X=(r,1.2), Y =(1,2,q) are two vectors. Find the values r and q such that both X and Y belong to H

Homework Equations

The Attempt at a Solution

To be honest, I don't know where to start. I'm studying this by myself and the book doesn't have an example similar to this question. All that i know is that [1,2,-1] is a null space of H. This would help me find the spanning family for H: I can write a in term of b and c i.e a =-2b+c. So (a,b,c) = (-2b+c,b,c) = b(-2,1,0)+c(1,0,1)..therefore (-2,1,0) and (1,0,1) form a spanning family for H. How can I find r and q then? I'd appreciate your help.
Thanks.

 

[Dick]

You are making this way too complicated. If X=(r,1,2) is an element of the subspace, then you want r+2*1-2=0, right? That's the definition of the subspace. And it's pretty easy to solve.

 

[mccoy1]

You are making this way too complicated. If X=(r,1,2) is an element of the subspace, then you want r+2*1-2=0, right? That's the definition of the subspace. And it's pretty easy to solve.

Thanks Dick.
So does that mean r+2-2 = 0, hence r=0 for vector X, 1+2*2-q=0, hence q =5 and for Y?

 

[Dick]

Thanks Dick.
So does that mean r+2-2 = 0, hence r=0 for vector X, 1+2*2-q=0, hence q =5 and for Y?

It certainly does.

 

[mccoy1]

It certainly does.

Thank you, it's appreciated. I was stuck on this for 2 good hours.