sertakan penjelasan nya ya, kalo bs yg panjang. ngaco report ya anj
Jawab:
[tex]\large\text{$\begin{aligned}&\boxed{\ \overrightarrow{BD}=\begin{pmatrix}\bf1\\\bf4\end{pmatrix}\ }\end{aligned}$}[/tex]
Penjelasan dengan langkah-langkah:
Vektor
Diketahui:
- Titik A(-3, 5), B(-1, 0) dan C(6, 2)
- Titik D berada pada ruas garis AC sehingga AD : CD = 1 : 2
Ditanya:
Vektor BD
Jawab:
AD : CD = 1 : 2
Perbandingan ini adalah perbandingan jarak, sehingga jika diurutkan, agar lebih mudah memahami vektornya, perbandingan tersebut menjadi:
AD : DC = 1 : 2, dengan titik D terletak pada ruas garis AC.
[tex]\large\text{$\begin{aligned}&\overrightarrow{AD}:\overrightarrow{DC}=\bf1:2\\\iff&\overrightarrow{DC}=2\overrightarrow{AD}\\\iff&\overrightarrow{OC}-\overrightarrow{OD}=2\left(\overrightarrow{OD}-\overrightarrow{OA}\right)\\\iff&\overrightarrow{OC}-\overrightarrow{OD}=2\overrightarrow{OD}-2\overrightarrow{OA}\\\iff&\overrightarrow{OC}+2\overrightarrow{OA}=2\overrightarrow{OD}+\overrightarrow{OD}\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}\iff&\overrightarrow{OC}+2\overrightarrow{OA}=3\overrightarrow{OD}\\\iff&\overrightarrow{OD}=\frac{1}{3}\left(\overrightarrow{OC}+2\overrightarrow{OA}\right)\end{aligned}$}[/tex]
Menentukan vektor BD
[tex]\large\text{$\begin{aligned}\overrightarrow{BD}&=\overrightarrow{OD}-\overrightarrow{OB}\\&=\frac{1}{3}\left(\overrightarrow{OC}+2\overrightarrow{OA}\right)-\overrightarrow{OB}\\&=\frac{1}{3}\left[\begin{pmatrix}6\\2\end{pmatrix}+2\begin{pmatrix}-3\\5\end{pmatrix}\right]-\begin{pmatrix}-1\\0\end{pmatrix}\\&=\frac{1}{3}\begin{pmatrix}6+(-6)\\2+10\end{pmatrix}-\begin{pmatrix}-1\\0\end{pmatrix}\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&=\frac{1}{3}\begin{pmatrix}0\\12\end{pmatrix}-\begin{pmatrix}-1\\0\end{pmatrix}\\&=\begin{pmatrix}0\\4\end{pmatrix}-\begin{pmatrix}-1\\0\end{pmatrix}\\&=\begin{pmatrix}0-(-1)\\4-0\end{pmatrix}\\\\\overrightarrow{BD}&=\begin{pmatrix}\bf1\\\bf4\end{pmatrix}\end{aligned}$}[/tex]
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