Diketahui vektor a=[−46] dan b=[2−1]. Proyeksi ortogonal vektor a pada vektor b adalah ...

Pertanyaan

Diketahui vektor $\overrightarrow{\mathrm a}=\begin{bmatrix}-4\\6\end{bmatrix}\;\mathrm{dan}\;\overrightarrow{\mathrm b}=\begin{bmatrix}2\\-1\end{bmatrix}$ . Proyeksi ortogonal vektor $\overrightarrow{\mathrm a}$ pada vektor $\overrightarrow b$ adalah ...

R. Risda

Master Teacher

Jawaban terverifikasi

Jawaban

Proyeksi ortogonal vektor straight a with rightwards arrow on top

pada vektor b with rightwards arrow on top

adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 14 over denominator 5 end fraction open square brackets table row 2 row cell negative 1 end cell end table close square brackets end cell end table$ .

Pembahasan

Diketahui :

straight a with rightwards arrow on top equals open square brackets table row cell negative 4 end cell row 6 end table close square brackets space dan space straight b with rightwards arrow on top equals open square brackets table row 2 row cell negative 1 end cell end table close square brackets

Ditanya :

Proyeksi ortogonal vektor straight a with rightwards arrow on top pada vektor b with rightwards arrow on top

Penyelesaian:
Perhatikan perhitungan berikut ini.

$\begin{array}{rcl}\left|\overrightarrow{\mathrm b}\right|^2\;&=&\left(\sqrt{2^2+1^2}\right)^2\\&=&4+1\\&=&5\end{array}$

Proyeksi ortogonal vektor $\begin{array}{rcl}&&\overrightarrow{\mathrm a}\end{array}$ pada vektor $\begin{array}{rcl}&&\overrightarrow{\mathrm b}\end{array}$ dapat diperoleh dengan perhitungan berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell straight a with rightwards arrow on top space pada space straight b with rightwards arrow on top space end cell equals cell fraction numerator straight a with rightwards arrow on top space straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar squared end fraction straight b with rightwards arrow on top end cell row cell open vertical bar straight b with rightwards arrow on top close vertical bar space end cell equals cell 2 squared plus 1 squared end cell row blank equals cell 4 plus 1 end cell row blank equals 5 row cell straight a with rightwards arrow on top space pada space straight b with rightwards arrow on top end cell equals cell fraction numerator open square brackets table row cell negative 4 end cell row 6 end table close square brackets open square brackets table row 2 row cell negative 1 end cell end table close square brackets over denominator 5 end fraction straight b with rightwards arrow on top end cell row cell straight a with rightwards arrow on top space pada space straight b with rightwards arrow on top end cell equals cell fraction numerator negative 8 minus 6 over denominator 5 end fraction straight b with rightwards arrow on top end cell row cell straight a with rightwards arrow on top space pada space straight b with rightwards arrow on top end cell equals cell fraction numerator negative 14 over denominator 5 end fraction straight b with rightwards arrow on top end cell row cell straight a with rightwards arrow on top space pada space straight b with rightwards arrow on top end cell equals cell fraction numerator negative 14 over denominator 5 end fraction open square brackets table row 2 row cell negative 1 end cell end table close square brackets end cell row blank blank blank end table$

Jadi, Proyeksi ortogonal vektor straight a with rightwards arrow on top pada vektor b with rightwards arrow on top adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 14 over denominator 5 end fraction open square brackets table row 2 row cell negative 1 end cell end table close square brackets end cell end table$ .