Contoh 9 Vektor

$\begin{array}{l}\\ 41.& \text{Diketahui segi empat ABCD dengan}\overrightarrow{DA}=\overrightarrow{a},\\ & \overrightarrow{DB}=\overrightarrow{b},\text{dan}\overrightarrow{DC}=\overrightarrow{c}.\text{Jika titik H pada AB}\\ & \text{dengan}\overrightarrow{AH}:\overrightarrow{HB}=1:2,\text{dan titik J pada BC}\\ & \text{dengan}\overrightarrow{BJ}:\overrightarrow{JC}=1:2\text{maka}\overrightarrow{HJ}=....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{-\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}}\\ \text{b}.{\displaystyle \frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}}\\ \text{c}.{\displaystyle \frac{-2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}}& \\ \text{d}.{\displaystyle \frac{-2\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}}{3}}\\ \text{e}.{\displaystyle \frac{-2\overrightarrow{a}+\overrightarrow{b}-2\overrightarrow{c}}{3}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Coba perhatikanlah ilustrasi berikut}\end{array}$

$.\begin{array}{rl}& \begin{array}{|cc|}\hline \text{Diketahui 1}& \text{Diketahui 2}\\ \begin{array}{rl}\overrightarrow{AH}:\overrightarrow{HB}& =1:2\\ \overrightarrow{h}& ={\displaystyle \frac{2\overrightarrow{a}+\overrightarrow{b}}{3}}\\ & \end{array}& \begin{array}{rl}\overrightarrow{BJ}:\overrightarrow{JC}& =1:2\\ \overrightarrow{j}& ={\displaystyle \frac{2\overrightarrow{b}+\overrightarrow{c}}{3}}\\ & \end{array}\\ \hline\end{array}\\ & \text{Proses Penyelesaian}\\ & \begin{array}{rl}\overrightarrow{HJ}& =\overrightarrow{j}-\overrightarrow{h}\\ & =\left({\displaystyle \frac{2\overrightarrow{b}+\overrightarrow{c}}{3}}\right)-\left({\displaystyle \frac{2\overrightarrow{a}+\overrightarrow{b}}{3}}\right)\\ & ={\displaystyle \frac{-2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}}\end{array}\end{array}$.

$\begin{array}{l}\\ 42.& \text{Supaya vektor}\overrightarrow{a}=\left(\begin{array}{c}x\\ 4\\ 7\end{array}\right),\text{dan}\overrightarrow{b}=\left(\begin{array}{c}6\\ y\\ 14\end{array}\right),\\ & \text{segaris, harga}x-y=....\\ & \begin{array}{l}\\ \text{a}.-5& & \text{d}.4\\ \text{b}.-2& \text{c}.3& \text{e}.6\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{Dikethui}& \text{bahwa}:\\ \text{vektor}& \overrightarrow{a}\text{dan}\overrightarrow{b}\text{segaris},\text{maka}\\ m\overrightarrow{a}& =\overrightarrow{b}\\ \text{dengan}& m\text{adalah skalar/faktor pengali}\\ m\left(\begin{array}{c}x\\ 4\\ 7\end{array}\right)& =\left(\begin{array}{c}6\\ y\\ 14\end{array}\right)\\ \text{di}\text{dapat}& \text{kan}\{\begin{array}{ll}mx& =6...................(1)\\ 4m& =y....................(2)\\ 7m& =14...................(3)\end{array}\\ \text{Dari per}& \text{samaan}(3)\text{akan}\\ \text{didapat}& \text{kan nilai}m=2\\ \text{maka}& \text{akan didapatkan juga}\{\begin{array}{ll}x& =3\\ y& =8\end{array}\\ \text{sehin}& \text{gga nilai dari}\\ x-y& =3-8\\ & =-5\end{array}\end{array}$.

$\begin{array}{l}\\ 43.& \text{Diketahui}O\text{titik pangkal}A(0,1,2)\\ & \text{dan}B(3,4,5),\text{maka luas segitiga}\\ & OAB\text{sama dengan}....\\ & \begin{array}{l}\\ \text{a}.3\sqrt{6}\\ \text{b}.{\displaystyle \frac{2}{3}\sqrt{6}}\\ \text{c}.{\displaystyle \frac{4}{3}\sqrt{6}}& \\ \text{d}.{\displaystyle \frac{3}{2}\sqrt{6}}\\ \text{e}.2\sqrt{6}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Coba perhatikanlah ilustrasi berikut}\\ & \begin{array}{rl}& \text{Misalkan luas segitiga}{\displaystyle \frac{1}{2}|\overrightarrow{p}\times \overrightarrow{q}|,}\\ & \text{dengan}\\ & \{\begin{array}{ll}\overrightarrow{p}& =\overline{OA}=\left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)\\ \overrightarrow{q}& =\overline{OB}=\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)\end{array}\end{array}\\ & \begin{array}{rl}\overrightarrow{p}& \times \overrightarrow{q}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ x_1& y_1& z_1\\ x_2& y_2& z_2\end{array}\right|\\ & =(y_1{z}_2-z_1{y}_2)\overrightarrow{i}-(x_1{z}_2-z_1{x}_2)\overrightarrow{j}+(x_1{y}_2-y_1{x}_2)\overrightarrow{k}\\ & =\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 0& 1& 2\\ 3& 4& 5\end{array}\right|\\ & =(5-8)\overrightarrow{i}-(0-6)\overrightarrow{j}+(0-3)\overrightarrow{k}\\ & =-3\overrightarrow{i}+6\overrightarrow{j}-3\overrightarrow{k}\\ & \text{Sehingga}\\ & |\overrightarrow{p}\times \overrightarrow{q}|=\sqrt{(-3{)}^2+6^2+(-3{)}^2}\\ & =\sqrt{9+36+9}=\sqrt{54}=3\sqrt{6}\\ & \text{Maka luas segi tiganya adalah}:\\ & \text{luas}\mathrm{△}ABC={\displaystyle \frac{1}{2}|\overrightarrow{p}\times \overrightarrow{q}|={\displaystyle \frac{1}{2}\left(3\sqrt{6}\right)={\displaystyle \frac{3}{2}\sqrt{6}}}}\end{array}\end{array}$.

$\begin{array}{l}\\ 44.& \text{Proyeksi skalar ortogonal}\\ & \overrightarrow{a}=2\overrightarrow{i}-3\overrightarrow{j}+6\overrightarrow{k},\text{pada}\\ & \overrightarrow{b}=\overrightarrow{i}+2\overrightarrow{j}+2\overrightarrow{k}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{4}{3}}& & & & & \text{d}.& {\displaystyle \frac{16}{3}}\\ \\ \text{b}.{\displaystyle \frac{8}{3}}& & \text{c}& {\displaystyle \frac{10}{3}}& & \text{e}.& {\displaystyle \frac{20}{3}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Diketahui bahwa}\\ & \{\begin{array}{ll}\overrightarrow{a}& =(2,-3,6)\\ \overrightarrow{b}& =(1,2,2)\end{array}\\ & \text{Selanjutnya}\\ & \begin{array}{r}\\ \left|\overrightarrow{c}\right|& =\left|{\displaystyle \frac{\overrightarrow{a}\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right|\\ & ={\displaystyle \frac{\left(\begin{array}{c}2\\ -3\\ 6\end{array}\right)\left(\begin{array}{c}1\\ 2\\ 2\end{array}\right)}{\sqrt{1^2+2^2+2^2}}}\\ & ={\displaystyle \frac{2-6+12}{\sqrt{9}}=\frac{8}{3}}\end{array}\end{array}$

$\begin{array}{l}\\ 45.& \text{Diketahui vektor}\overrightarrow{a}=3\overrightarrow{i}-4\overrightarrow{j}+p\overrightarrow{k}\\ & \text{dan}\overrightarrow{b}=2\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}.\text{Jika panjang}\\ & \text{proyeksi vektor}\overrightarrow{a}\text{pada}\overrightarrow{b}\text{adalah}\\ & {\displaystyle \frac{4}{\sqrt{17}},\text{maka nilai}p\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle -2}& & & & & \text{d}.& {\displaystyle 2}\\ \\ \text{b}.{\displaystyle -1}& & \text{c}& {\displaystyle 1}& & \text{e}.& {\displaystyle 3}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Panjang proyeksi skalar vektor}\overrightarrow{a}\text{pada}\overrightarrow{b}\\ & \begin{array}{r}\\ \left|\overrightarrow{c}\right|& ={\displaystyle \frac{\overrightarrow{a}\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\\ {\displaystyle \frac{4}{\sqrt{17}}}& ={\displaystyle \frac{\left(\begin{array}{c}3\\ -4\\ p\end{array}\right)\left(\begin{array}{c}2\\ 2\\ -3\end{array}\right)}{\sqrt{2^2+2^2+(-3{)}^2}}}\\ {\displaystyle \frac{4}{\sqrt{17}}}& ={\displaystyle \frac{6-8-3p}{\sqrt{17}}}\\ 4& =-2-3p\\ 6& =-3p\\ -3p& =6\\ p& =-2\end{array}\end{array}$