Calc 3 question! Perpendicular surfaces

[sergnio]

Show that the surfaces given by xy^5 + yz^5 + zx^5 = 3 and x^2 + y^2 = 2z^2 intersect in a perpendicular manner at (1,1,1).
I've been staring at this problem for over an hour, looking online, through my book, through my notes as to how to go about doing this and I have noo idea.
Thanks in advance!

 

[Deleted member 4993]

Show that the surfaces given by xy^5 + yz^5 + zx^5 = 3 and x^2 + y^2 = 2z^2 intersect in a perpendicular manner at (1,1,1).
I've been staring at this problem for over an hour, looking online, through my book, through my notes as to how to go about doing this and I have noo idea.
Thanks in advance!

What are the normal and gradients of these surfaces at (1,1,1)?

 

[sergnio]

Hm perhaps that's the first step to solving the problem.. (those are not given)
I tried finding the gradient, but I don't believe I calculated it properly.

 

[sergnio]

What are the normal and gradients of these surfaces at (1,1,1)?.

The gradient to the first problem is {y^5 + 5 x^4 z, 5xy^4 + z^5, x^5 + 5t z^4}
and the gradient to the second is {2x, 2y, -4z}

 

[HallsofIvy]

Hm perhaps that's the first step to solving the problem.. (those are not given)
I tried finding the gradient, but I don't believe I calculated it properly.

Then tell us what you did and what you got.

 

[DrPhil]

The gradient to the first problem is {y^5 + 5 x^4 z, 5xy^4 + z^5, x^5 + 5t z^4}
and the gradient to the second is {2x, 2y, -4z}

Evaluate those two vectors at the point (1,1,1).

Are they perpendicular to each other at that point?