We're given vector function r(t)= t i - t^2 j - t^3 k, how woud u lfind [r'(1)] X [r''(1)] . (cross product) ?

r(t) = (t, -t^2, -t^3)

r'(t) = (1 -2t, -3t^2)

r''(t) = (0, -2, -6t)

r'(1) = (1, -2, -3)

r''(1) = (0, -2, -6)

r'(1) x r"(1) = det( (i, j, k) (1, - 2, -3)(0, -2, -6) )

= 6i +6j - 2k.

r' = i - 2t j - 3t^2 k

r'' = 0 i - 2 j - 6 t k

r'(1) = i - 2 j - 3 k

r''(1) = 0 i - 2 j - 6 k

Cross product

a × b = (a2b3 − a3b2) i + (a3b1 − a1b3) j + (a1b2 − a2b1) k = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1)

(1,-2,-3)X(0,-2,-6) = ((-6*-2)-(-3*-2)) i + ((-3*0) - (1*-6)) j + ((1*-2)-(-2*0)) k = (12-6)i + (0+6)j + (-2-0)k

= 6i + 6j -2k