What is the principal unit normal vector to the curve at the specified value of the parameter?

Question

#r(t)=sqrt(2)thati+e^thatj+e^-thatk; t=0#

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Answer 1 · 2018-01-02T01:37:41

#hatn = sqrt2/2hati+ 1/2hatj-1/2hatk#

The normal vector at any point on the curve is the gradient of the curve evaluated at the specified parametric value.

Compute the gradient:

#gradr(t) = (del(sqrt2t))/(del t)hati+ (del(e^t))/(del t)hatj+(del(e^-t))/(del t)hatk#

#gradr(t) = sqrt2hati+ e^thatj-e^-thatk#

A normal vector, #vecn#, is the gradient evaluated at #t=0#

#vecn = sqrt2hati+ hatj-hatk#

We make it the unit vector by dividing by its magnitude:

#hatn = (sqrt2hati+ hatj-hatk)/sqrt((sqrt2)^2+ 1^2+ (-1)^2)#

#hatn = (sqrt2hati+ hatj-hatk)/sqrt4#

#hatn = sqrt2/2hati+ 1/2hatj-1/2hatk#