2.diketahui koordinat titik K(1, -5) Y(5, 2) dan T(4, 6) jika YK=NT, tentukan koordinat titik N
3.diketahui vektor p=i+2j-3k, q=3i+5k, r=2i-4j+k dan z=2p+q-r tentukan nilai vektor z
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Jawaban:
- Koordinat titik H adalah (19, 12).
- Koordinat titik N adalah (8, 13).
- [tex]\overrightarrow{z}&=3\overrightarrow{i}+8\overrightarrow{j}-2\overrightarrow{k}[/tex]
Pembahasan
Vektor
Nomor 1
[tex]\large\text{$\begin{aligned}\overrightarrow{NH}&=(13,8)\,,\ \textsf{titik }N(6,4)\\\\\overrightarrow{NH}&=\overrightarrow{OH}-\overrightarrow{ON}\\\Leftrightarrow\overrightarrow{OH}&=\overrightarrow{NH}+\overrightarrow{ON}\\&=(13,8)+(6,4)\\&=(13+6,\ 8+4)\\&=(19,12)\end{aligned}$}[/tex]
∴ Berdasarkan vektor posisi OH, koordinat titik H adalah (19, 12).
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Nomor 2
[tex]\large\text{$\begin{aligned}&K(1,-5),\ Y(5, 2),\ T(4, 6)\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&\overrightarrow{YK}=\overrightarrow{NT}\\&{\iff}\overrightarrow{OK}-\overrightarrow{OY}=\overrightarrow{OT}-\overrightarrow{ON}\\&{\iff}\overrightarrow{ON}=\overrightarrow{OT}-\left(\overrightarrow{OK}-\overrightarrow{OY}\right)\\&{\qquad\quad\ \ \:\:}=\overrightarrow{OT}-\overrightarrow{OK}+\overrightarrow{OY}\\&{\iff}\overrightarrow{ON}=(4, 6)-(1,-5)+(5, 2)\\&{\qquad\quad\ \ \:\:}=(4-1+5,\ 6-(-5)+2)\\&{\qquad\quad\ \ \:\:}=(8, 13)\end{aligned}$}[/tex]
∴ Berdasarkan vektor posisi ON, koordinat titik N adalah (8, 13).
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Nomor 3
[tex]\large\text{$\begin{aligned}&\begin{cases}\overrightarrow{p}=\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}\\\\\overrightarrow{q}=3\overrightarrow{i}+5\overrightarrow{k}\\\\\overrightarrow{r}=2\overrightarrow{i}-4\overrightarrow{j}+\overrightarrow{k}\\\\\overrightarrow{z}=2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r}\end{cases}\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}\overrightarrow{z}&=2\left(\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}\right)\\&\ \:\vdots\quad+3\overrightarrow{i}+5\overrightarrow{k}\\&\ \:\vdots\quad-\left(2\overrightarrow{i}-4\overrightarrow{j}+\overrightarrow{k}\right)\\\\&=2\overrightarrow{i}+4\overrightarrow{j}-6\overrightarrow{k}\\&\ \:\vdots\quad+3\overrightarrow{i}+0\overrightarrow{j}+5\overrightarrow{k}\\&\ \:\vdots\quad-2\overrightarrow{i}+4\overrightarrow{j}-\overrightarrow{k}\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}\overrightarrow{z}&=(2+3-2)\overrightarrow{i}\\&\ \:\vdots\quad+(4+0+4)\overrightarrow{j}\\&\ \:\vdots\quad+(-6+5-1)\overrightarrow{k}\\\\&=3\overrightarrow{i}+8\overrightarrow{j}+\left(-2\overrightarrow{k}\right)\\\\\overrightarrow{z}&=3\overrightarrow{i}+8\overrightarrow{j}-2\overrightarrow{k}\end{aligned}$}[/tex]
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