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)]

a. 22

b. 70

c. 166

d. 184

e. 246

or is it none of these?

Answer 1

The derivative is a linear operator, so the derivative of a vector is the vector of its derivatives. so

r'(t) = 1i-2tj-3t^2k
and
r''(t) = 0i-2j-6tk

The dot product is the sum of the product of the corresponding entries. if

A = a_1i+a_2j+a_3k
and
B = b_1i+b_2j+b_3k
then
dot(A,B) = a_1*b_1 + a_2*b_2 + a_3*b_3

you can do the rest.

Answer 2

Given the vector function r(t) = t i - t² j - t³ k, how would you find [r'(1)] · [r''(1)].

The vector r is a function of t.

r'(t) = i - 2t j - 3t² k
r''(t) = 0i - 2j - 6t k

[r'(t)] • [r''(t)] = <1, -2t, -3t²> • <0, -2, -6t> = 0 + 4t + 18t²

[r'(1)] • [r''(1)] = 0 + 4*1 + 18*1² = 22

The answer is a.