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!
Best Answer
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.