Question

Num paralelogramo ABCD sabe-se que A (1,3,–2) e que as diagonais são =(4,2,–3) e =(–2,0,1).Calcule as coordenadas dos outros três vértices.

Answer 1

ABCD
A = (1,3,-2)
Diagonal AC = (4,2,-3)
Diagonal BD = (-2,0,1)

AC = C - A
C = AC + A
C = (4,2,-3) + (1,3,-2)
C = (5,5,-5)

Por ser um paralelogramo, os lados opostos são congruentes e paralelos.

Portanto AB = DC e BC = AD
Por serem vetores, sabemos que:

AC = AB + BC
e
BD = AD - AB

Se somarmos AC e BC, teremos:

AC + BD = AB + BC + AD - AB
AC + BD = BC + AD

Como BC = AD

(4,2,-3) + (-2,0,1) = 2 . BC
(2,2,-2) = 2BC
BC = (1,1,-1)

Sabemos que:

BC = C - B
(1,1,-1) = (5,5,-5) - B
B = (5,5,-5) - (1,1,-1)
B = (4,4,-4)

Se BC =AD e AD = D - A

(1,1,-1) = D - (1,3,-2)
D = (1,1,-1) + (1,3,-2)
D = (2,4,-3)

Resumindo:
$\boxed {A = (1,3,-2)\  B = (4,4,-4)\  C = (5,5,-5)\  D = (2,4,-3)\ }$

=)