Question

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

Answer 1

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

Explanation:

`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)`