Diketahui titik-titik A(1, −3, 2), B(1, 3, 10) dan C(4, −1, 8). Jika D merupakan proyeksi titik C pada garis AB maka AD sama dengan ...

Pertanyaan

Diketahui titik-titik $\text{A}\left(1,\;-3,\;2\right)$ , $\text{B}\left(1,\;3,\;10\right)$ dan $\text{C}\left(4,\;-1,\;8\right)$ . Jika $\text{D}$ merupakan proyeksi titik $\text{C}$ pada garis $\text{AB}$ maka $\overrightarrow{\text{AD}}$ sama dengan ...

H. Eka

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

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Pembahasan

Panjang vektor a with rightwards arrow on top equals x i with rightwards arrow on top plus y j with rightwards arrow on top plus z k with rightwards arrow on top , yaitu open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared plus z squared end root

Jika vektor a with rightwards arrow on top equals a subscript 1 i with rightwards arrow on top plus a subscript 2 j with rightwards arrow on top plus a subscript 3 k with rightwards arrow on top dan b with rightwards arrow on top equals b subscript 1 i with rightwards arrow on top plus b subscript 2 j with rightwards arrow on top plus b subscript 3 k with rightwards arrow on top , maka

a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 times b subscript 1 plus a subscript 2 times b subscript 2 plus a subscript 3 times b subscript 3

Proyeksi vektor orthogonal vektor a with rightwards arrow on top pada b with rightwards arrow on top dapat ditentukan menggunakan rumus berikut.

$c with rightwards arrow on top equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction times b with rightwards arrow on top$

Komponen vektor stack A C with rightwards arrow on top dan stack A B with rightwards arrow on top dapat ditentukan sebagai berikut.

Proyeksi vektor stack A C with rightwards harpoon with barb upwards on top pada stack A B with rightwards arrow on top adalah sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell stack A D with rightwards arrow on top end cell equals cell fraction numerator stack A C with rightwards arrow on top times stack A B with rightwards arrow on top over denominator open vertical bar stack A B with rightwards arrow on top close vertical bar squared end fraction times stack A B with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 3 row 2 row 6 end table close parentheses open parentheses table row 0 row 6 row 8 end table close parentheses over denominator open parentheses square root of 0 squared plus 6 squared plus 8 squared end root close parentheses squared end fraction times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell fraction numerator 3 times 0 plus 2 times 6 plus 6 times 8 over denominator open parentheses square root of 100 close parentheses squared end fraction times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell 60 over 100 times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell 3 over 5 times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell times open parentheses table row 0 row cell 18 over 5 end cell row cell 24 over 5 end cell end table close parentheses end cell end table$

Diperoleh vektor stack A D with rightwards arrow on top equals 18 over 5 j with rightwards arrow on top plus 24 over 5 k with rightwards arrow on top

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