Given the vector function r(t) = t i - t^2 j - t^3 k, how would you find [r'(1)] · [r''(1)] . (dot product)

Question

possible solutions for [r'(1)] · [r''(1)] (dot product)

a. 22

b. 70

c. 166

d. 184

e. 246

or is it none of these?

Answer 1

r'(1) = i - 2j - 3k
r"(1) = -2j - 6k
[r'(1)] · [r''(1)] = 4 + 18 = 22

Answer 2

r(t) = t i - t^2 j - t^3 k
r'(t) = i - 2t j - 3t^2 k ....(1)
r''(t) = - 2j -6t k ........(2)

Putting t = 1 in (1) and (2) and taking the dot product gives:
(1,-2, -3).(0, -2, -6)
= 0 + 4 + 18
= 22.
Answer (a).

Answer 3

r `(t) = -2t j - 3t² k
r`(1) = - 2j - 3k

r"(t) = - 2j - 6t k
r"(1) = - 2j - 6k

r`(1) . r"(1) = 4 + 18 = 22

OPTION a)