How do I find the normal vector to the plane (x, y, z) = (1, 2, 3) + t(2, 3, 4) + s(4, 5, 6) , it goes through origo..?
$\displaystyle r(s,t)=(1+2t+4s,2+3t+5s,3+4t+6s)$
the normal vector is given by
$\displaystyle \frac{\partial r}{\partial t} \times \frac{\partial r}{\partial s} $
$\displaystyle \begin{vmatrix}
\vec i && \vec j && \vec k \\
2 && 3 && 4 \\
4 && 5&& 6 \\
\end{vmatrix}=(18-20)\vec i - (12-16) \vec j + (10-12) \vec k =-2 \vec i +4 \vec j -2 \vec k
$