Answer to Question #223671 in Chemical Engineering for Lokika

Question #223671
Find unit normal vector to the surface xy^{2}+2yz=8 at the point (3,-2,1)
1
Expert's answer
2021-08-05T18:05:47-0400

Given:

"F(x,y,z)=xy^2+2yz-8,\\quad M(3,-2,1)"

The unit normal vector to the surface "F(x,y,z)=0" is given by

"{\\bf N}=\\frac{\\nabla F}{|\\nabla F|}"

"\\nabla F=\\left({\\bf i}\\frac{\\partial}{\\partial x}+{\\bf j}\\frac{\\partial}{\\partial y}+{\\bf k}\\frac{\\partial}{\\partial z}\\right)(xy^2+2yz-8)"

"=\\left({\\bf i}y^2+{\\bf j}(2xy+2z)+{\\bf k}2y\\right)=4{\\bf i}-10{\\bf j}-4{\\bf k}"

"|\\nabla F|=\\sqrt{4^2+(-10)^2+(-4)^2}=2\\sqrt{33}"

Finally

"{\\bf N}=\\frac{2{\\bf i}-5{\\bf j}-2{\\bf k}}{\\sqrt{33}}"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS