yup, I know that (U dot V) is = 1 because U dot V = (u1 * v1 + u2 * v2 +...+uN * vN)
I think it see the problem more clearly now: Is it that we create a vector x and dot multiply it by both vectors U and V and then pick values that will give the dot product between X and V and X and V = 0 for both cases?
In this case x_1 =x_2 = -x_3 works to make both dot products zero so we can create X = (1, 1, -1).
If this is the correct procedure, is there a general way of creating an orthogonal vector, or is it done by inspection?
Hello, kingsolomonsgrave!
$\displaystyle \text{Find a unit vector orthogonal to both }\vec u= \langle1,1,0\rangle\text{ and }\vec v = \langle0,1,1\rangle.$
Why isn't this basic theorem being used?
A vector $\displaystyle \vec n$ orthogonal to $\displaystyle \vec u$ and $\displaystyle \vec v$ is given by: .$\displaystyle \vec u \times \vec v$
Hence: .$\displaystyle \vec n \;=\;\begin{vmatrix} i&j&k \\ 1&1&0 \\ 0&1&1\end{vmatrix} \;=\;i(1-0) - j(1-0) + k(1-0) \;=\;i - j + k \;=\;\langle1,\text{-}1,1\rangle $
A unit vector is: .$\displaystyle \vec U_1 \:=\:\left\langle \tfrac{1}{\sqrt{3}},\:\tfrac{-1}{\sqrt{3}},\:\tfrac{1}{\sqrt{3}}\right\rangle$
Another is its negative: .$\displaystyle \vec U_2 \:=\:\left\langle \tfrac{-1}{\sqrt{3}},\:\tfrac{1}{\sqrt{3}},\:\tfrac{-1}{\sqrt{3}}\right\rangle $