Question #223671

Find unit normal vector to the surface xy^{2}+2yz=8 at the point (3,-2,1)

Expert's answer

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}}"

Learn more about our help with Assignments: Chemical Engineering

## Comments

## Leave a comment