Find the principal unit normal vector N for the curve r(t).
-r(t) = (10 sin t)i + (10 cos t)j + 9k
A) N = (sin t)i + (cos t)j
B) N = (-sin t)i + (cos t)j
C) N = (-sin t)i - (cos t)j
D) N = (-cos t)i - (sin t)j

Question

Quizplus · Accepted Answer

The formula for the principal unit normal vector N for a curve is N = T' / ||T'||, where T is the tangent vector and ' denotes differentiation with respect to t.

r(t) = (10 sin t)i + (10 cos t)j + 9k
T(t) = r'(t) = 10 cos t)i - 10 sin t)j
||T|| = sqrt((10 cos t)^2 + (-10 sin t)^2) = 10
T'(t) = (-10 sin t)i - (10 cos t)j
||T'|| = sqrt(((-10 sin t)^2) + ((-10 cos t)^2)) = 10

Therefore, N(t) = T'(t) / ||T'|| = [(-10 sin t)/10]i + [(10 cos t)/10]j = (-sin t)i + (cos t)j

The correct answer is (C) N = (-sin t)i - (cos t)j.

Find the Principal Unit Normal Vector N for the Curve