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

Question

possible book solutions for [r'(1)] X [r''(1)] (cross product).....

a. 6 i - 6 j + 2 k

b. 6 i + 6 j - 2 k

c. 12 i - 16 j + 6 k

d. 18 i - 36 j + 4 k

e. 24 i - 96 j + 18 k

f. 24 i - 64 j + 48 k

or is it none of these?

.

Answer 1

find r ' (t), and r ''(t). Cross them.

Cross product of a 3 space is basically the determinant matrix for the two vectors
l i j kl
l r'(t) i r ' (t) j r' (t) kl
l r'' (t) i r '' (t) j r'' (t) kl
and you expand this.

you get: (-12 + 6) i - (-6 - 0) j + (2 - 0) k

Answer 2

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

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

r`(1) X r"(1) =
|i---j---k|
|1 -2 -3|
|0 -2 -6|

= 6i + 6j - 2k

ANSWER b.