This is the question from my book,
If $p=2i-j+3k$, $q=5i+2j$ and $r=4i+j+k$, find a set of numbers $f$, $g$ and $h$ such that $fp+gq+hr=0$. What does this tell you about the translations represented by $p$, $q$ and $r$?
So, how am I going to find out $f$, $g$ and $h$? I just don't know where to start. Please help me with the second part of the question too. Thanks, in advance, to anyone who helps me out with this question.
Hint: I think that what the book means by $i$,$j$ and $k$ is the canonical vectors $ i = (1,0,0)$, $j=(0,1,0)$ and $k=(0,0,1)$. Sometimes they are written as $\hat{i}$, $\hat{j}$ and $\hat{k}$ to avoid possible confusion with scalar quantities (I think this is the case).
So, in fact: $$p=(2,-1,3)$$ $$q= (5,1,0)$$ $$r=(4,1,1)$$
So the system you want to solve is:
$$ 2f + 5g + 4h = 0$$ $$ -f + g + h = 0$$ $$ 3f + h = 0$$
Can you continue from here?