Для решения данной задачи воспользуемся методом координат. Предположим, что точка A находится в начале координат (0,0,0). Тогда координаты остальных точек будут следующими:
B(3,0,0)
C(0,4,0)
D(0,0,5)
M(1.5,0,0)
N(1.5,2,0)
I(1.5,2,2.5)
K(1.5,2,-2.5)
Теперь можем найти координаты векторов AB, BC, BD, MN, BN, NK, CB, BA, DB, NC, KN.
AB = B – A = (3, 0, 0) – (0, 0, 0) = (3, 0, 0) = 3i
BC = C – B = (0, 4, 0) – (3, 0, 0) = (-3, 4, 0) = -3i + 4j
BD = D – B = (0, 0, 5) – (3, 0, 0) = (-3, 0, 5) = -3i + 5k
MN = N – M = (1.5, 2, 0) – (1.5, 0, 0) = (0, 2, 0) = 2j
BN = N – B = (1.5, 2, 0) – (3, 0, 0) = (-1.5, 2, 0) = -1.5i + 2j
NK = K – N = (1.5, 2, -2.5) – (1.5, 2, 0) = (0, 0, -2.5) = -2.5k
CB = B – C = (3, 0, 0) – (0, 4, 0) = (3, -4, 0) = 3i – 4j
BA = A – B = (0, 0, 0) – (3, 0, 0) = (-3, 0, 0) = -3i
DB = B – D = (3, 0, 0) – (0, 0, 5) = (3, 0, -5) = 3i – 5k
NC = C – N = (0, 4, 0) – (1.5, 2, 0) = (-1.5, 2, 0) = -1.5i + 2j
KN = N – K = (1.5, 2, 0) – (1.5, 2, -2.5) = (0, 0, 2.5) = 2.5k
Теперь можем посчитать длины этих векторов:
|AB| = sqrt(3^2 + 0^2 + 0^2) = sqrt(9) = 3
|BC| = sqrt((-3)^2 + 4^2 + 0^2) = sqrt(9 + 16) = sqrt(25) = 5
|BD| = sqrt((-3)^2 + 0^2 + 5^2) = sqrt(9 + 25) = sqrt(34)
|MN| = sqrt(0^2 + 2^2 + 0^2) = sqrt(4) = 2
|BN| = sqrt((-1.5)^2 + 2^2 + 0^2) = sqrt(2.25 + 4) = sqrt(6.25) = 2.5
|NK| = sqrt(0^2 + 0^2 + (-2.5)^2) = sqrt(6.25) = 2.5
|CB| = sqrt(3^2 + (-4)^2 + 0^2) = sqrt(9 + 16) = sqrt(25) = 5
|BA| = sqrt((-3)^2 + 0^2 + 0^2) = sqrt(9) = 3
|DB| = sqrt(3^2 + 0^2 + (-5)^2) = sqrt(9 + 25) = sqrt(34)
|NC| = sqrt((-1.5)^2 + 2^2 + 0^2) = sqrt(2.25 + 4) = sqrt(6.25) = 2.5
|KN| = sqrt(0^2 + 0^2 + 2.5^2) = sqrt(6.25) = 2.5
Таким образом, найдены длины векторов AB = 3, BC = 5, BD = sqrt(34), MN = 2, BN = 2.5, NK = 2.5, CB = 5, BA = 3, DB = sqrt(34), NC = 2.5, KN = 2.5.