Contoh 1 Vektor

$\begin{array}{l}\\ 1.& \text{Perhatikanlah ilustrasi gambar berikut}\end{array}$

$\begin{array}{l}\\ .& \text{maka vektor}\overrightarrow{u}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 3\overrightarrow{i}+5\overrightarrow{j}& & & \text{d}.& -3\overrightarrow{i}-5\overrightarrow{j}\\ \text{b}.& 5\overrightarrow{i}+3\overrightarrow{j}& \text{c}.& -3\overrightarrow{i}+5\overrightarrow{j}& \text{e}.& -5\overrightarrow{i}+3\overrightarrow{j}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \text{Kita perhatikan lagi gambarnya}\\ & \begin{array}{rl}& \text{Vektor}\overrightarrow{u}\text{jika dinyatakan sebagai }\\ & \text{kombinasi linear adalah}\overrightarrow{u}=-3\overrightarrow{i}+5\overrightarrow{j}\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Panjang vektor}\overrightarrow{p}=\left(\begin{array}{c}4\\ -8\end{array}\right)\text{adalah}....\\ & \text{a}.{\displaystyle \sqrt{4}}\\ & \text{b}.\sqrt{12}\\ & \text{b}.\sqrt{20}\\ & \text{d}.{\displaystyle \sqrt{80}}\\ & \text{e}.\sqrt{100}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overrightarrow{p}& =\left(\begin{array}{c}4\\ -8\end{array}\right),\text{maka besar dari vektor}\\ \overrightarrow{p}& \text{adalah}=\left|\overrightarrow{p}\right|=\sqrt{x^2+y^2}\\ & \text{Yaitu}\\ \left|\overrightarrow{p}\right|& =\sqrt{4^2+(-8{)}^2}\\ & =\sqrt{16+64}=\sqrt{80}\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Perhatikanlah gambar berikut}\\ & \end{array}$.

$\begin{array}{l}\\ .& \text{Panjang vektor}\overrightarrow{h}\text{tersebut di atas adalah}....\\ & \text{a}.5\\ & \text{b}.7\\ & \text{c}.10\\ & \text{d}.12\\ & \text{e}.15\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\text{Diketahui}& \overrightarrow{h}=\overline{OH}=8\overrightarrow{i}+6\overrightarrow{j}\\ \overrightarrow{h}& =\sqrt{x_H^2+y_H^2}\\ & =\sqrt{8^2+6^2}\mathbf{\text{(ingat tripel Pythagoras)}}\\ & =\sqrt{{10}^2}=10\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Vektor satuan dari}\overrightarrow{q}=3\overrightarrow{i}-4\overrightarrow{j}\text{adalah}....\\ & \text{a}.{\displaystyle \frac{4}{5}\overrightarrow{i}-\frac{3}{5}\overrightarrow{j}}\\ & \text{b}.{\displaystyle \frac{3}{5}\overrightarrow{i}-\frac{4}{5}\overrightarrow{j}}\\ & \text{c}.3\overrightarrow{i}-4\overrightarrow{j}\\ & \text{d}.4\overrightarrow{i}-3\overrightarrow{j}\\ & \text{e}.15\overrightarrow{i}-20\overrightarrow{j}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overrightarrow{q}& =3\overrightarrow{i}-4\overrightarrow{j}\\ & \text{Vektor satuan dari vektor}\overrightarrow{q}\text{adalah}:\\ & {\hat{e}}_{{}_{\overrightarrow{q}}}={\displaystyle \frac{1}{\left|\overrightarrow{q}\right|}.\overrightarrow{q}}\\ & \text{Sehingga}\\ {\hat{e}}_{{}_{\overrightarrow{q}}}& ={\displaystyle \frac{1}{\sqrt{3^2+(-4{)}^2}}.\left(\begin{array}{c}3\\ -4\end{array}\right)}\\ & ={\displaystyle \frac{1}{5}\left(\begin{array}{c}3\\ -4\end{array}\right)}\\ & \text{atau dalam vektor basis}\\ & ={\displaystyle \frac{3}{5}\overrightarrow{i}-\frac{4}{5}\overrightarrow{j}}\end{array}\end{array}$

$\begin{array}{l}\\ 5.& \text{Vektor berikut yang memiliki panjang}\\ & 29\text{satuan adalah}....\\ & \text{a}.{\displaystyle 18\overrightarrow{i}-19\overrightarrow{j}}\\ & \text{b}.{\displaystyle 19\overrightarrow{i}-20\overrightarrow{j}}\\ & \text{c}.20\overrightarrow{i}-21\overrightarrow{j}\\ & \text{d}.21\overrightarrow{i}-22\overrightarrow{j}\\ & \text{e}.22\overrightarrow{i}-23\overrightarrow{j}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\text{Ingat}& \text{lah akan tigaan Pythagoras}\\ & \{\begin{array}{ll}(3,4,5)& \Rightarrow 3^2+4^2=5^2\\ (5,12,13)& \Rightarrow 5^2+{12}^2={13}^2\\ (8,15,17)& \Rightarrow \cdots \\ (20,21,29)& \Rightarrow \cdots \\ ⋮& dll\end{array}\\ \text{Sehin}& \text{gga}\\ \text{yang}& \text{paling mungkin adalah}:\\ & =\sqrt{{20}^2+{21}^2}=\sqrt{400+441}\\ & =\sqrt{841}\\ & =29\end{array}\end{array}$