Diberikan vektor a = ( -2 p 2√2) dengan p € Real dan vektor b = ( 1 1 √2 ). Jika a dan b membentuk sudut 60°, maka kosinus sudut antara vektor a dan a + b adala

Mapel : Matematika
Kelas : X SMA
Bab : Vektor

Pembahasan :
a . b = |a| |b| Cos 60°
[-2 p 2√2] [1 1 √2] = √(p² + 12) √4 (1/2)
-2 + p + 4 = √(p² + 12)
P + 2 = √(p² + 12)
(P + 2)² = [√(p² + 12)]²
P² + 4p + 4 = p² + 12
4p = 12 - 4
4p = 8
P = 2

a + b = [-2 2 2√2] + [1 1 √2]
a + b = [-1 3 3√2]

Maka...
a . (a + b) = |a| |a + b| cos β
[-2 2 2√2] [-1 3 3√2] = √16 √28 Cos β
2 + 6 + 12 = 4√28 Cos β
20 = 4√(7.4) Cos β
20 = 8√7 Cos β
Cos β = 20/(8√7)
Cos β = 5/(2√7)
Cos β = 5/(2√7) x (√7)/(√7)
Cos β = (5√7)/14