Contoh 5 Soal dan Pembahasan Materi Vektor

  $\begin{array}{l}\\ 21.& \text{Jika}\overrightarrow{g}=\left(\begin{array}{c}{3}^{x+y}\\ 5\end{array}\right)\text{dan}\overrightarrow{h}=\left(\begin{array}{c}81\\ {\displaystyle \frac{y+7}{2}}\end{array}\right)\\ & \text{sehingga}\overrightarrow{g}=\overrightarrow{h}\text{nilai dari}4x-3y=....\\ & \text{a}.-5\\ & \text{b}.-1\\ & \text{c}.0\\ & \text{d}.5\\ & \text{e}.10\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\text{Diketahui}& \text{bahwa}:\overrightarrow{g}=\overrightarrow{h}\\ \left(\begin{array}{c}{3}^{x+y}\\ 5\end{array}\right)& =\left(\begin{array}{c}81\\ {\displaystyle \frac{y+7}{2}}\end{array}\right)\\ 3^{x+y}& =81=3^4\iff x+y=4\\ {\displaystyle \frac{y+7}{2}}& =5\iff y=10-7=3,\text{sehingga}\\ x+y& =4\iff x+3=4\iff x=4-3=1,\\ & \text{maka}\\ 4x-3y& =4(1)-3(3)\\ & =4-9=-5\end{array}\end{array}$

$\begin{array}{l}\\ 22.& \text{Vektor}\overrightarrow{m}=\left(\begin{array}{c}-2\\ 5\end{array}\right)\text{searah }\\ & \text{dengan vektor}....\\ & \text{a}.{\displaystyle \left(\begin{array}{c}2\\ -5\end{array}\right)}\\ & \text{b}.{\displaystyle \left(\begin{array}{c}2\\ 5\end{array}\right)}\\ & \text{c}.{\displaystyle \left(\begin{array}{c}-6\\ 15\end{array}\right)}\\ & \text{d}.{\displaystyle \left(\begin{array}{c}-4\\ 5\end{array}\right)}\\ & \text{e}.{\displaystyle \left(\begin{array}{c}-3\\ 10\end{array}\right)}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}& \text{Vektor}\overrightarrow{m}\text{searah dengan vektor}k.\overrightarrow{m}\\ & k.\overrightarrow{m}=k\left(\begin{array}{c}-2\\ 5\end{array}\right),\text{dengan}k\text{skalar positif}\\ & \begin{array}{|cc|}\hline \text{a}& \text{b}\\ \left(\begin{array}{c}2\\ -5\end{array}\right)=-\left(\begin{array}{c}-2\\ 5\end{array}\right)& \left(\begin{array}{c}2\\ 5\end{array}\right)=...\\ \text{c}& \text{d}\\ \left(\begin{array}{c}-6\\ 15\end{array}\right)=3\left(\begin{array}{c}-2\\ 5\end{array}\right)& \left(\begin{array}{c}-4\\ 5\end{array}\right)=...\\ \text{e}& \\ \left(\begin{array}{c}-3\\ 10\end{array}\right)=...& \\ \hline\end{array}\end{array}\end{array}$

$\begin{array}{l}\\ 23.& \text{Jika vektor}\overline{AC}=\overrightarrow{p},\overline{BC}=\overrightarrow{q}\text{dan}\\ & \overline{AD}:\overline{DC}=1:2,\text{maka vektor}\\ & \overline{BD}\text{bila dinyatakan}\\ & \text{dalam}\overrightarrow{p}\text{dan}\overrightarrow{q}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{3}(3\overrightarrow{p}-2\overrightarrow{q})}\\ \text{b}.{\displaystyle \frac{1}{3}(3\overrightarrow{q}-2\overrightarrow{p})}\\ \text{c}.{\displaystyle \frac{1}{3}(\overrightarrow{p}-2\overrightarrow{q})}\\ \text{d}.{\displaystyle (\overrightarrow{p}-{\displaystyle \frac{1}{3}\overrightarrow{q}})}\\ \text{e}.{\displaystyle \frac{1}{3}(\overrightarrow{p}-\overrightarrow{q})}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\overline{AC}:\overline{CD}& =3:-2\\ \overline{CD}& =-{\displaystyle \frac{2}{3}\overline{AC}}\\ & ={\displaystyle -\frac{2}{3}\overrightarrow{p}}\\ \text{maka},& \\ \overline{BD}& =\overline{BC}+\overline{CD}\\ & =\overrightarrow{q}+(-{\displaystyle \frac{2}{3}\overrightarrow{p}})\\ & ={\displaystyle \frac{1}{3}(3\overrightarrow{q}-2\overrightarrow{p})}\end{array}\end{array}$

$.\text{Gambar berikut untuk soal 24}$

$\begin{array}{l}\\ 24.& \text{Jika vektor}\overline{AC}=\overrightarrow{p},\overline{BC}=\overrightarrow{q}\text{dan}\\ & \overline{AD}:\overline{DC}=1:2,\text{maka vektor}\overline{BD}\\ & \text{bila dinyatakan dalam}\overrightarrow{p}\text{dan}\overrightarrow{q}\text{adalah}....\\ & \text{a}.{\displaystyle \frac{1}{3}(3\overrightarrow{p}-2\overrightarrow{q})}\\ & \text{b}.{\displaystyle \frac{1}{3}(3\overrightarrow{q}-2\overrightarrow{p})}\\ & \text{c}.{\displaystyle (\overrightarrow{p}-{\displaystyle \frac{1}{3}\overrightarrow{q}})}\\ & \text{d}.{\displaystyle \frac{1}{3}(\overrightarrow{p}-2\overrightarrow{q})}\\ & \text{e}.{\displaystyle \frac{1}{3}(\overrightarrow{p}-\overrightarrow{q})}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overline{AC}:\overline{CD}& =3:-2\\ \overline{CD}& =-{\displaystyle \frac{2}{3}\overline{AC}}\\ & ={\displaystyle -\frac{2}{3}\overrightarrow{p}}\\ \text{maka},& \\ \overline{BD}& =\overline{BC}+\overline{CD}\\ & =\overrightarrow{q}+(-{\displaystyle \frac{2}{3}\overrightarrow{p}})\\ & ={\displaystyle \frac{1}{3}(3\overrightarrow{q}-2\overrightarrow{p})}\end{array}\end{array}$

$\begin{array}{l}\\ 25.& \text{Jika titik}A(2,6)\text{dan}B(5,3)\text{demikian }\\ & \text{juga titik}P\text{terletak pada}\overline{AB}\\ & \text{dengan}\overline{AP}:\overline{PB}=2:1,\text{maka vektor }\\ & \text{posisi}\overrightarrow{p}\text{adalah}....\\ & \text{a}.\left(\begin{array}{c}4\\ 4\end{array}\right)\\ & \text{b}.\left(\begin{array}{c}4\\ 5\end{array}\right)\\ & \text{c}.\left(\begin{array}{c}-4\\ 4\end{array}\right)\\ & \text{d}.\left(\begin{array}{c}4\\ 2\end{array}\right)\\ & \text{e}.\left(\begin{array}{c}-4\\ 6\end{array}\right)\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overline{AP}:\overline{PB}& =2:1\\ \overline{AP}& =2\overline{PB}\\ \overrightarrow{p}-\overrightarrow{a}& =2(\overrightarrow{b}-\overrightarrow{p})\\ \overrightarrow{p}+2\overrightarrow{p}& =\overrightarrow{a}+2\overrightarrow{b}\\ 3\overrightarrow{p}& =\overrightarrow{a}+2\overrightarrow{b}\\ \overrightarrow{p}& ={\displaystyle \frac{1}{3}(\overrightarrow{a}+2\overrightarrow{b})}\\ & ={\displaystyle \frac{1}{3}\left(\begin{array}{c}2+2.5\\ 6+2.3\end{array}\right)}\\ & ={\displaystyle \frac{1}{3}\left(\begin{array}{c}12\\ 12\end{array}\right)}\\ & =\left(\begin{array}{c}4\\ 4\end{array}\right)\end{array}\end{array}$