Question

from the "advanced example" in the lecture clip on lagrange multipliers. Why are u1, u2 and u3 perpendicular to the sides of the base triangle??

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-41-advanced-example/

Answer 1

I guess it is hard to visualize.
But consider the "slant height" of a face from point P down to its base leg. That slant height makes a 90 degree angle with the leg


now rotate that slant height until it lies in the x-y plane

it will remain 90 degrees to the leg

Also, point P when projected down to the x-y plane will lie on this rotated "slant height"
Thus we have (looking down on the base)