Diberikan kubus ABCD.EFGH dengan panjang 4 satuan. a.

a. ||BG|| = 4√2, ||AG|| = 4√3; b. CD = (-4,0,0), FC = (0,4,-4), EC = (4,4,-4); c. (0,0,0)

Pembahasan

Diberikan kubus ABCD.EFGH dengan panjang rusuk 4 satuan.

a. Menghitung panjang vektor BG dan AG:
Misalkan titik A berada di (0,0,0).
Maka:
B = (4,0,0)
C = (4,4,0)
D = (0,4,0)
E = (0,0,4)
F = (4,0,4)
G = (4,4,4)
H = (0,4,4)

Vektor BG = G - B = (4,4,4) - (4,0,0) = (0,4,4)
Panjang vektor BG = ||BG|| = sqrt(0^2 + 4^2 + 4^2) = sqrt(0 + 16 + 16) = sqrt(32) = 4 * sqrt(2)

Vektor AG = G - A = (4,4,4) - (0,0,0) = (4,4,4)
Panjang vektor AG = ||AG|| = sqrt(4^2 + 4^2 + 4^2) = sqrt(16 + 16 + 16) = sqrt(48) = 4 * sqrt(3)

b. Menentukan vektor kolom atau baris dari vektor CD, FC, dan EC:
Vektor CD = D - C = (0,4,0) - (4,4,0) = (-4,0,0)
Vektor FC = C - F = (4,4,0) - (4,0,4) = (0,4,-4)
Vektor EC = C - E = (4,4,0) - (0,0,4) = (4,4,-4)

c. Menentukan hasil operasi vektor dari vektor CD - vektor FC + vektor EC:
CD - FC + EC = (-4,0,0) - (0,4,-4) + (4,4,-4)
= (-4 - 0 + 4, 0 - 4 + 4, 0 - (-4) + (-4))
= (0, 0, 0)