find a third vector to form a set that spans $\mathbb R^{3}$

Question

i have two vectors that i need to combine with a third unknown vector to create a spanning set, but i can't figure out how to do this. i have written the two vectors i have below. this is from a singular value decomposition problem.

$u_1 = (\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$

$u_2 = (\frac{-1}{3\sqrt{2}}, \frac{-1}{3\sqrt{2}}, \frac{-4}{3\sqrt{2}})$

i am trying to solve a problem based on an example in my e-text, but they kind of skipped over this part and i can't figure out how to get this similar result for the vectors i'm working with. i have $u_1$ and $u_2$, but i can't figure out how to get $u_3$.

any advice? thank you in advance!

Answer 1

Since $u_1$ and $u_2$ as given are unit vectors that are perpendicular to each other

(their dot product is $0$),

their vector cross product will be a unit vector perpendicular to them, which is what is sought.

Answer 2

For any set of $n - k$ linearly independent vectors in an $n$-dimensional vector space, we can extend to a basis by appending some $k$ of the standard basis vectors $e_1, \ldots, e_n$ (proof: add all of them in and sift).

In this case, it's easy to see that $(0,0,1)$ is linearly independent of your set of vectors, so that one will do.