The problem itself is formulated as such:
Find 2 orthonormal vectors x,y so that span{x,y} = span{(0,1,1), (1,2,3)}
My understanding is:
span{(0,1,1), (1,2,3)} is a plane. Instead of finding 2 orthogonal vectors generating said subspace (plane), I can choose one of the vectors (f.e. (0,1,1)) and only need to find another vector from that plane, that is orthogonal to it (0,1,1). (Although I don't know how to do this)
I don't know if my understanding is correct, how could I solve this problem?
Best Answer
In general we have that given $v_1$ and $v_2$ we can construct $u_2$ orthogonal to $v_1$ by
$$u_2=v_2-\left(v_2 \cdot \frac{v_1}{|v_1|}\right) \frac{v_1}{|v_1|}$$
according to Gram–Schmidt process idea indeed
$$u_2 \cdot v_1=v_2\cdot v_1-\left(v_2 \cdot \frac{v_1}{|v_1|}\right) \frac{v_1\cdot v_1}{|v_1|}=v_2\cdot v_1-v_2\cdot v_1=0$$