Question

Determine m e n para que se tenha

(a) (m, n, 2) × (4, −1, 3) = (8, −1, −11)

(b) (m, n, 2) × (1, 1, −1) = (0, 0, 0)​

Answer 1

Resposta:

(a) (m, n, 2) × (4, −1, 3) = (8, −1, −11)

Produto vetorial

x      y     z       x     y

m    n    2        m    n

4     -1     3       4     -1

det=3nx +8y -mz -3my  +2x  -4nz

det=[ x*(3n+2)   ; y*(8-3m)  ; z*(-m-4n)] = (8x,-1,-11z)

{ (3n+2) = 8 ==>n=2

{ 8-3m=-1  ==>m=3

{ -m-4n=-11   ==>  -3-8 =-11 ==>  OK

n=2  e m=3

(b) (m, n, 2) × (1, 1, −1) = (0, 0, 0)​

x      y     z       x     y

m    n    2        m    n

1      1    -1       1     -1

det= x*(-n) +y*2  +z*(-m) +y *m +x*(-2) +z*n == (0*x+0*y+0*z)

det= x*(-n-2) +y*(2+m)  +z*(-m+n)  = (0*x+0*y+0*z)

-n-2=0 ==>n=-2

2+m=0 ==>m=-2

-m+n=0 ==>-(-2)-2 =0  ==> OK

n=-2  e n=-2