Diketahui |vektor a|=6 dan |vektor b|=8, serta sudut antara

-288

Pembahasan

Untuk menghitung nilai dari (3 vektor a - 2 vektor b).(3 vektor b + 2 vektor a):

Diketahui:
*   |vektor a| = 6
*   |vektor b| = 8
*   Sudut antara vektor a dan b (θ) = 120°

Kita akan menggunakan sifat perkalian titik (dot product) dan identitas aljabar.

Langkah-langkah:

1.  **Jabarkan perkalian titik:**
    (3 vektor a - 2 vektor b).(3 vektor b + 2 vektor a)
    = (3a ⋅ 3b) + (3a ⋅ 2a) - (2b ⋅ 3b) - (2b ⋅ 2a)
    = 9(a ⋅ b) + 6(a ⋅ a) - 6(b ⋅ b) - 4(b ⋅ a)

2.  **Gunakan sifat perkalian titik:**
    *   a ⋅ a = |vektor a|^2
    *   b ⋅ b = |vektor b|^2
    *   a ⋅ b = b ⋅ a
    *   a ⋅ b = |vektor a| |vektor b| cos(θ)

3.  **Substitusikan nilai-nilai yang diketahui:**
    *   a ⋅ a = |vektor a|^2 = 6^2 = 36
    *   b ⋅ b = |vektor b|^2 = 8^2 = 64
    *   a ⋅ b = |vektor a| |vektor b| cos(120°)
        cos(120°) = -1/2
        a ⋅ b = (6)(8)(-1/2) = 48 * (-1/2) = -24

4.  **Hitung hasil penjabaran:**
    = 9(a ⋅ b) + 6(a ⋅ a) - 6(b ⋅ b) - 4(b ⋅ a)
    = 9(-24) + 6(36) - 6(64) - 4(-24)
    = -216 + 216 - 384 + 96
    = 0 - 384 + 96
    = -288

Jadi, nilai dari (3 vektor a - 2 vektor b).(3 vektor b + 2 vektor a) adalah -288.