Surface Normal

**weasley74** · April 9th 2008, 12:45 PM

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..?

**TheEmptySet** · April 9th 2008, 12:56 PM

Originally Posted by weasley74

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..?

$r(s,t)=(1+2t+4s,2+3t+5s,3+4t+6s)$

the normal vector is given by

$\frac{\partial r}{\partial t} \times \frac{\partial r}{\partial s}$

$\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 $

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