I am quite confused with this question. The question itself wasn't originally in English, so I'll try my best to translate it:
"Complete the following linearly independent families with vectors of $\mathbb R$$^3$ in a basis of $\mathbb R$$^3$."
So, is the question asking for a family of 3 vectors, since the basis is OF $\mathbb R$$^3$? I guess the source of my confusion is finding out whether they are referring to the dimensions of the vectors within the family rather than the dimension of the vector space itself.
Assuming that the family does indeed comprise 3 vectors, let's say that the unfinished family given is ((1,1,1)). What is the best way to go about this problem? Should I just look for some vectors that would make it linearly indepedent? I feel like that might not work with more complicated problems.
My best idea is to to find ($u_1$,$u_2$,$u_3$), ($v_1$,$v_2$,$v_3$) $\in$ $\mathbb R$$^3$, such that for any a,b,c $\in$ $\mathbb R$ we have a(1,1,1) + b($u_1$,$u_2$,$u_3$) + c($v_1$,$v_2$,$v_3$) = (0,0,0) $\implies$ a = 0, b = 0, c = 0.
We would have:
a + b$u_1$ + c$v_1$ = 0
a + b$u_2$ + c$v_2$ = 0
a + b$u_3$ + c$v_3$ = 0
And now, I just have to find the values for u and v which would imply that all the scalars are nul. It sounds like it should be really simple, but I'm not sure how to go about this. Any help is greatly appreciated. Thank you.
We can find a linearly independent orthogonal vector to the given vectors according to one of the two following ways
find the third vector by cross product $v_3=v_1\times v_2$
find an orthogonal vector by dot product solving the system $(a,b,c)\cdot v_1=0$ and $(a,b,c)\cdot v_2=0$