Contoh 4 Soal dan Pembahasan Materi Vektor

 $\begin{array}{l}\\ 16.& \text{Diketahui titik A(-1,1,0) dan titik B(1,-2,2)}\\ & \text{maka panjang vektor}\overrightarrow{BA}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& \sqrt{2}& & & \text{d}.& \sqrt{17}\\ \text{b}.& \sqrt{5}& \text{c}.& \sqrt{9}& \text{e}.& \sqrt{21}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}& \text{Diketahui}\text{sebagaimana pada soal}\\ & \text{maka pan}\text{jang vektor}\overrightarrow{BA}\text{adalah}\\ & \left|\overrightarrow{BA}\right|=\sqrt{(1-(-1){)}^2+(-2-1{)}^2+(2-0{)}^2}\\ & =\sqrt{2^2+(-3{)}^2+2^2}\\ & =\sqrt{4+9+4}\\ & =\sqrt{17}\end{array}\end{array}$

$\begin{array}{l}\\ 17.& \text{Vektor satuan untuk vektor}\overrightarrow{a}=\left(\begin{array}{ccc}2,& 1,& -2\end{array}\right)=....\\ & \begin{array}{l}\\ \text{a}.& \left(\begin{array}{ccc}-{\displaystyle \frac{2}{3},}& -{\displaystyle \frac{1}{3},}& {\displaystyle \frac{2}{3}}\end{array}\right)& \\ \\ \text{b}.& \left(\begin{array}{ccc}{\displaystyle \frac{2}{3},}& {\displaystyle \frac{1}{3},}& -{\displaystyle \frac{2}{3}}\end{array}\right)& \\ \\ \text{c}.& \left(\begin{array}{ccc}{\displaystyle \frac{2}{4},}& {\displaystyle \frac{1}{4},}& -{\displaystyle \frac{2}{4}}\end{array}\right)& \\ \\ \text{d}.& \left(\begin{array}{ccc}-{\displaystyle \frac{2}{4},}& -{\displaystyle \frac{1}{4},}& {\displaystyle \frac{2}{4}}\end{array}\right)\\ \\ \text{e}.& \left(\begin{array}{ccc}-{\displaystyle \frac{2}{9},}& -{\displaystyle \frac{1}{9},}& {\displaystyle \frac{2}{9}}\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\text{Vektor }& \text{satuan dari vektor}\overrightarrow{a}\text{adalah}:\\ \hat{a}& ={\displaystyle \frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}}\\ & ={\displaystyle \frac{\left(\begin{array}{ccc}2,& 1,& -2\end{array}\right)}{\sqrt{2^2+1^2+(-2{)}^2}}}\\ & ={\displaystyle \frac{\left(\begin{array}{ccc}2,& 1,& -2\end{array}\right)}{\sqrt{9}}}\\ & ={\displaystyle \frac{\left(\begin{array}{ccc}2,& 1,& -2\end{array}\right)}{3}}\\ & =\left(\begin{array}{ccc}{\displaystyle \frac{2}{3},}& {\displaystyle \frac{1}{3},}& -{\displaystyle \frac{2}{3}}\end{array}\right)\end{array}\end{array}$

$\begin{array}{l}\\ 18.& \text{Jika titik A(-2,3,5) dan B(4,1,-3)},\\ & \text{maka vektor posisi AB adalah}....\\ & \begin{array}{l}\\ \text{a}.& \left(\begin{array}{ccc}-6,& 2,& 8\end{array}\right)& \\ \text{b}.& \left(\begin{array}{ccc}8,& 2,& -6\end{array}\right)& \\ \text{c}.& \left(\begin{array}{ccc}6,& -2,& -8\end{array}\right)& \\ \text{d}.& \left(\begin{array}{ccc}-8& -2,& 6\end{array}\right)\\ \text{e}.& \left(\begin{array}{ccc}2,& 4,& 2\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\text{Vektor posisi}& \text{dari}\overrightarrow{AB}\text{adalah}:\\ \overrightarrow{AB}& =\overrightarrow{OB}-\overrightarrow{OA}\\ & =\left(\begin{array}{c}4\\ 1\\ -3\end{array}\right)-\left(\begin{array}{c}-2\\ 3\\ 5\end{array}\right)\\ & =\left(\begin{array}{c}4+2\\ 1-3\\ -3-5\end{array}\right)\\ & =\left(\begin{array}{c}6\\ -2\\ -8\end{array}\right)\mathbf{\text{atau}}\\ & =\left(\begin{array}{ccc}6,& -2,& -8\end{array}\right)\end{array}\end{array}$.

$\begin{array}{l}\\ 19.& \text{Jika}\overrightarrow{p}=\left(\begin{array}{c}{}^2\log 8x\\ {({}^2\log x)}^y\end{array}\right)\text{dan}\overrightarrow{q}=\left(\begin{array}{c}5\\ 8\end{array}\right)\\ & \text{sehingga}\overrightarrow{p}=\overrightarrow{q}\text{nilai dari}x.y=....\\ & \text{a}.6\\ & \text{b}.12\\ & \text{c}.18\\ & \text{d}.24\\ & \text{e}.30\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\text{Diketahui}& \text{bahwa}:\overrightarrow{p}=\overrightarrow{q}\\ \left(\begin{array}{c}{}^2\log 8x\\ {({}^2\log x)}^y\end{array}\right)& =\left(\begin{array}{c}5\\ 8\end{array}\right),\text{maka}\\ 8x& =2^5=32\\ \iff x& ={\displaystyle \frac{32}{8}=4}\\ {({}^2\log 4)}^y& =8\\ \iff 2^y& =8=2^3\\ \iff y& =3\\ \text{Sehingga}& \\ x.y& =4\times 3=12\end{array}\end{array}$

$\begin{array}{l}\\ 20.& \text{Jika}\overrightarrow{p}=\left(\begin{array}{c}3x\\ 4x+y\end{array}\right)\text{dan}\overrightarrow{q}=\left(\begin{array}{c}{\displaystyle \frac{2x-4}{2}}\\ 6\end{array}\right)\\ & \text{sehingga}\overrightarrow{p}=\overrightarrow{q}\text{nilai dari}2x+y=....\\ & \text{a}.-12\\ & \text{b}.0\\ & \text{c}.8\\ & \text{d}.9\\ & \text{e}.19\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\text{Diketahui}& \text{bahwa}:\overrightarrow{p}=\overrightarrow{q}\\ \left(\begin{array}{c}3x\\ 4x+y\end{array}\right)& =\left(\begin{array}{c}{\displaystyle \frac{2x-4}{2}}\\ 6\end{array}\right)\\ 3x& ={\displaystyle \frac{2x-4}{2}\iff 6x=2x-4}\\ \iff x& =-1\\ 4(-1)+y& =6\iff -4+y=6\\ \iff y& =6+4\\ y& =10\\ x+y& =(-1)+10=9\end{array}\end{array}$