Consider the surface xyz=30. How do you find the unit normal vector to the surface at (2,5,3)?

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Answer 1 · 2016-12-29T11:23:38

$hatvecn=1/19(15hatveci+6hatvecj+10hatveck)$

$xyz=30$

write as

$f(x,y,z)=xyz-30=0$

vector normal to $" "f(x)" "$ is given by $gradf(x,y,z)$

$gradf(x,y,z)=(hatvecidel/(delx)+hatvecjdel/(dely)+hatveckdel/(delz))(xyz-30)$

$gradf(x,y,z)=yzhatveci+xzhatvecj+xyhatveck$

$gradf(2,5,3)=(5xx3)hatveci+(2xx3)hatvecj+(2xx5)hatveck$

$gradf(2,5,3)=15hatveci+6hatvecj+10hatveck$

call this normal vector $""vecn$

unit vector in this direction is given by

$hatvecn=vecn/|vecn|$

$|vecn|=sqrt(15^2+6^2+10^2)=19$

$hatvecn=1/19(15hatveci+6hatvecj+10hatveck)$