Contoh
Diketahui vektor-vektor a (0, – 2, – 1), b (2, 3, 4), dan c (– 3, 0, 3),
tentukan:
1. a + b
2. b + a
3. b + c
4. b – c
5. c – b
6. a + a
7. a – a
8. a + o
9. (a + b) + c
10. a + (b + c)
Jawab:
1. a + b = (0, – 2, – 1) + (2,
3, 4) = (0 + 2, – 2 + 3, – 1 + 4) = ( 2, 1, 3)
Jadi, a + b = (2, 1, 3).
2. b + a = (2, 3, 4) + (0, –
2, – 1) = (2 + 0, 3 + (– 2), 4 + (– 1)) = ( 2, 1, 3)
Jadi, b + a = (2, 1, 3).
3. b + c = (2, 3, 4) + (–3, 0, 3) =
(2 + (– 3), 3 + 0, 4 + 3) = (–1, 3, 7)
Jadi, b + c = (– 1, 3 7).
4. b – c = (2, 3, 4) – (– 3,
0, 3) = (2 – (– 3), 3 – 0, 4 – 3) = ( 5, 3, 1)
Jadi, b – c = (5, 3, 1).
5. c – b = (– 3, 0, 3) – (2, 3, 4) =
(– 3 – 2, 0 – 3, 3 – 4) = (–5, –3, –1)
Jadi, c – b = (–5, –3, –1).
6. a + a = (0, –2, –1) + (0, –2,
–1) = ((0 + 0, –2 + (–2), –1 + (–1)) = (0, –4, –2)
Jadi, a + a = (0, –4, –1).
7. a – a = (0, –2, –1) – (0, –2,
–1) = ((0 + 0, –2 – (–2), –1 – (–1)) = (0, 0, 0) = o
Jadi, a – a = o.
8. a + o = (0, –2, –1) + (0,
0, 0) = (0 + 0, –2 + 0, –1 + 0) = (0, –2, –1) = a
Jadi, a + o = a.
9. (a + b) + c = (2, 3, 4) + (–3, 0,
3) = (2 + (–3), 1 + 0, 3 + 3) = (–1, 1, 6)
Jadi, (a + b) + c = (–1,
1, 6).
10. a + (b + c) = (0,
–2, –1) + (–1, 3, 7) = (0 + (–1), –2 + 3, –1 + 7) = (–1, 1, 6)
Jadi, a + (b + c) = (–1,
1, 6).
Contoh
1. Diketahui vektor a = (1, 4,
5) dan b = (2, 3, 2), tentukan vector c = 2a + 3b.
Jawab:
c = 2a + 3b = 2(1, 4, 5) +
3(2, 3, 2)
= (2 × 1, 2 × 4, 2 × 5) + (3 × 2, 3 × 3, 3 ×
2)
= (2, 8, 10) + (6, 9, 6)
= (8, 17, 16)
Jadi, c = 2a + 3b = (8, 17,
16).
2. Buktikan bahwa vektor u (– 3, 0,
6) sejajar dengan vector v (1, 0, - 2).
Bukti:
Untuk
membuktikan bahwa vektor u = (–3, 0, 6)
sejajar dengan vektor v = (1, 0, –2), kalian
harus menunjukkan ada bilangan real k sehingga u = k v.
u = k v ⇒
u = k v = o
(–3, 0, 6) - k(1, 0, –2) = (0, 0, 0)
(–3, 0, 6) – (k, 0,
–2k) = (0, 0, 0)
(–3 – k, 0, 6 + 2k) = (0, 0,
0)
Didapat, k
= –3, maka, u = –3v.
Jadi, vektor u
(–3, 0, 6) sejajar dengan vektor v (1, 0, –2).
Sumber
Labels:
Matematika
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