Question

Sejam os vetores u=3i+2j+6k, v=-4i-2j+k e w=(2u+3v)-(5k).o produto misto (u,v,w) é

Answer 1

Boa noite Neiva

Sejam os vetores

u = 3i + 2j + 6k
v = -4i -  2j+ k
w = (2u + 3v) - (5k)

w = 2*(3i + 2j + 6k) + 3*(-4i - 2j + k) - 5k
w = 6i + 4j + 12k - 12i - 6j + 3k - 5k
w = -6i - 2j + 10k 

produto misto 

(  3     2    6 )   3    2
( -4    -2    1 )  -4   -2
( -6    -2   10 ) -6   -2

det = -60 - 12 + 48 - 72 + 6 + 80 = -10 

Answer 2

$(u,v,w)= {detA \ | \ A= \left[\begin{array}{ccc}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1&w_2&w_3\end{array}\right]$

$u=3i+2j+6k \\ (u_1,u_2,u_3)=(3,2,6)\\ \\ v=-4i-2j+k\\(v_1,v_2,v_3)=(-4,-2,1)\\\\ w=(2u+3v)-(5k) \\ w=(2[3i+2j+6k]+3[-4i-2j+k])-(5k) \\ w=(6i+4j+12k-12i-6j+3k)-(5k) \\ w=-6i-2j+15k-5k \\w=-6i-2j+10k \\ (w_1,w_2,w_3)=(-6,-2,10)\\ \\ Logo: \\\\ A= \left[\begin{array}{ccc}3&2&6\\-4&-2&1\\-6&-2&10\end{array}\right]$

$detA=3.(-2).10+2.1.(-6)+6.(-4).(-2)\\-6.(-2).(-6)-3.1.(-2)-2.(-4).10 \\\\ detA= -60-12+48-72+6+80 \\\\ detA=-10 \\\\ (u,v,w)=detA \\ \\ (u,v,w)=-10$

Resposta: Alternativa E.