Would anyone know how to begin a problem where a function z(x,y) is given and x(t),y(t),z(t).
Thanks!
the normal line to the surface is expressed as:
N$\displaystyle = \displaystyle\frac{\partial{z}}{\partial{x}}$$\displaystyle \space$i$\displaystyle + \displaystyle\frac{\partial{z}}{\partial{y}}$$\displaystyle \space$j$\displaystyle - $$\displaystyle \space$k
$\displaystyle \displaystyle\frac{\partial{z}}{\partial{x}} = y$; $\displaystyle \space$ $\displaystyle \displaystyle\frac{\partial{z}}{\partial{y}} = x$
N$\displaystyle = y$$\displaystyle \space$i$\displaystyle + x$$\displaystyle \space$j$\displaystyle - $$\displaystyle \space$k
from the given line we have the symmetric equations:
$\displaystyle \displaystyle\frac{x - 3}{-2} = \displaystyle\frac{y - 4}{5} = \displaystyle\frac{z - 3}{3}$
which gives us:
$\displaystyle \space$ $\displaystyle -2$i$\displaystyle \space$$\displaystyle +$$\displaystyle \space$$\displaystyle 5$j$\displaystyle \space$$\displaystyle +$$\displaystyle \space$$\displaystyle 3$k
what this is saying is that the given line coincides with vector:
$\displaystyle -2$$\displaystyle \space$i$\displaystyle + 5$$\displaystyle \space$j$\displaystyle + 3$$\displaystyle \space$k
the normal line to the surface coincides with the vector:
$\displaystyle y$$\displaystyle \space$i$\displaystyle + x$$\displaystyle \space$j$\displaystyle - $k
since the two vectors are parallel, that would make the two lines parallel. So then,
$\displaystyle y$$\displaystyle \space$i$\displaystyle + x$$\displaystyle \space$j$\displaystyle - $k$\displaystyle = \displaystyle\frac{2}{3}$$\displaystyle \space$i$\displaystyle - \displaystyle\frac{5}{3}$$\displaystyle \space$j$\displaystyle - $$\displaystyle \space$k