f(t) = t * i + 2 * t² * j + 3 * t³ * k

g(t) = 2t i + j - 3t^2 k, t ≤ 0

Passo 1

g) Fala, galera!

Primeiro vamos calcular vec f (t) xx vec g (t) para adiantar a nossa vida:

vec f (t) xx vec g (t) = (t vec i + 2 t^2 vec j + 3 t^3 vec k) xx (2 t vec i + vec j - 3 t^2 vec k)

= [[[vec i,vec j,vec k],[t,2,t^2,3,t^3],[2,t,1,-,3,t^2]]]

= (2 t^2 * (- 3 t^2)) vec i + (3 t^3 * 2 t) vec j + (t * 1) vec k - (2 t * 2 t^2) vec k - (1 * 3 t^3) vec i - (- 3 t^2 * t) vec j

= - 6 t^4 vec i + 6 t^4 vec j + t vec k - 3 t^3 vec i + 3 t^3 vec j - 4 t^3 vec k

vec f (t) xx vec g (t) = (- 6 t^4 - 3 t^3) vec i + (6 t^4 + 3 t^3) vec j + (t - 4 t^3) vec k

Passo 2

underset(t -> 0)(lim) [ vec f (t) xx vec g ( t ) ]

Ufa! Tá acabando!

Já calculamos vec f (t) xx vec g ( t ) no Passo anterior, agora vamos aplicar o limite, mas repare que t -> 0 :

underset(t -> 0)(lim) [vec f (t) xx vec g (t)] = underset(t -> 0)(lim) (- 6 t^4 - 3 t^3) vec i + underset(t -> 0)(lim) (6 t^4 + 3 t^3) vec j + underset(t -> 0)(lim) (t - 4 t^3) vec k

= (- 0 - 0) vec i + (0 + 0) vec j + (0 - 0) vec k

underset(t -> 0)(lim) [vec f (t) xx vec g (t)] = 0 vec i + 0 vec j + 0 vec k

Resposta

g) underset(t -> 0)(lim) [vec f (t) xx vec g (t)] = 0 vec i + 0 vec j + 0 vec k