Contoh Soal Vektor dan Pembahasannya

Diketahui tiga buah vektor:
A = i + 2j - k
B = 4i + 2j + 3k
C = 2j - 3k

Tentukan: 
a. A ∙ ( B × C )
b. A ∙ ( B + C )
c. A × ( B + C )

Pembahasan:

a.  B × C
= ( 4i + 2j + 3k ) × ( 2j -3k )
= ( 4i × 2j ) + (4i × (-3k)) + ( 2j × 2j ) + (2j × (-3k)) + ( 3k × 2j ) + (3k × (-3k))
= 8k + 12j - 6i - 6i
= -12i + 12j + 8k

A ∙ ( B × C )
= ( i + 2j - k ) ∙ ( -12i + 12j + 8k )
= ( 1 ∙ (-12)) + ( 2 ∙ 12 ) + ((-1) ∙ (8))
= -12 + 24 - 8
= 4

b.  B + C
= ( 4i + 2j + 3k ) + ( 2j -3k )
= 4i + 4j

A ∙ ( B + C )
= ( i + 2j - k ) ∙  ( 4i + 4j )
= ( 1 ∙ 4 ) + ( 2 ∙ 4 ) + ((-1) ∙ 0 )
= 4 + 8
= 12

c. A × ( B + C )
= ( i + 2j - k ) × ( 4i + 4j )
= ( i ∙ 4i ) + ( i ∙ 4j ) + ( 2j ∙ 4i ) + ( 2j ∙ 4j ) + ((-k) ∙ 4i) + ((-k) ∙ 4j)
= 4k - 8k - 4j + 4i
= 4i - 4j - 4k