Vector translation question

If p = 2i-j+3k, q=5i+2j and r=4i+j+k, find a set of numbers f,g and h such that fp +gp +hr = 0. What does this tell you about the translations represented by p,q and r?

I thought to do this I just need to solve 3 simultaneous equations,

1. 2f + 5g +4h =0

2. -f + 2g +h =0

3. 3f + h =0

But how as I have no constant value and just solve each variable to be zero.

The book gives 1,2,-3 (or multiples of) and I can see that this is right but I dont know how to get it.

Also, what does this tell you about the translations represented by p,q and r??

Thanks.