Tentukan proyeksi ortogonal AB terhadap AC. Jika A (1,1,2) B (-1,0,3) C (2,-3,4)

vektor

A = (1, 1, 2)
B = (-1, 0, 3)
C = (2, -3, 4)

proyeksi orthogonal AB terhadap AC


maka
[tex] \\ |c| = | \frac{ab.ac}{ { |ac| }^{2} } | ac \\ \\ [/tex]
cari AB dan AC

AB = b - a = (-1, 0, 3) - (1, 1, 2) = (-2, -1, 1)
AC = c - a = (2, -3, 4) - (1, 1, 2) = (1, -4, 2)

maka AB × AC =
(-2, -1, 1) × (1, -4, 2) = (-2 + 4 + 2) = 4

panjang |AC|²
= (1)² + (-4)² + (2)²
= 1 + 16 + 4
= 21

masukan ke rumus sehingga diperoleh proyeksi orthogonalnya

[tex] |c| = | \frac{4}{21} | (1. \: \: \: \: \: \: \: - 4. \: \: \: \: \: \: \: \: \: 2) \\ \\ atau \: \: |c| = ( \frac{4}{21} \: \: \: \: \: \: \: - \frac{ 16}{21} \: \: \: \: \: \: \: \: \: \frac{8}{21} )[/tex]