VectorValued Functions..find the principle unit normal vector to the curve given below at the specified point.

Question

r (t) = t i + (4/t) j , t = 2

Answer 1

r(t) = (t, 4/t)
r'(t) = (1, -4/t²)
||r'(t)|| = √(1+16/t⁴)

T(t) = r'(t)/||r'(t)|| = (1/√(1+16/t⁴), -4/√(t⁴+16))
T'(t) = (-1/2 (1+16/t⁴)^(-3/2) * (-64/t⁵), 2 (t⁴+16)^(-3/2) * (4t³))

T'(2) = (-1/2 (1+1)^(-3/2) * (-64/32), 2 (16+16)^(-3/2) * 32) = (1/(2√2), 64/(32√32)) = (1/(2√2), 1/(2√2))

||T'(2)|| = √(1/8 + 1/8) = 1/2

N(2) = T'(2)/||T'(2)|| = (1/√2, 1/√2)