Contoh 3 Soal dan Pembahasan Materi Vektor

 $\begin{array}{l}\\ 11.& \text{Jika vektor}\overrightarrow{a}=\left(\begin{array}{c}6\\ -4\end{array}\right)\text{dan}\overrightarrow{b}=\left(\begin{array}{c}3\\ 2\end{array}\right),\\ & \text{maka}3\overrightarrow{a}-2\overrightarrow{b}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.\left(\begin{array}{c}12\\ -16\end{array}\right)& & \text{d}.\left(\begin{array}{c}24\\ 16\end{array}\right)\\ \text{b}.\left(\begin{array}{c}24\\ -16\end{array}\right)& \text{c}.\left(\begin{array}{c}12\\ 16\end{array}\right)& \text{e}.\left(\begin{array}{c}-12\\ -16\end{array}\right)\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}3\overrightarrow{a}-2\overrightarrow{b}& =3\left(\begin{array}{c}6\\ -4\end{array}\right)-2\left(\begin{array}{c}3\\ 2\end{array}\right)\\ & =\left(\begin{array}{c}18-6\\ -12-4\end{array}\right)\\ & =\left(\begin{array}{c}12\\ -16\end{array}\right)\end{array}\end{array}$

$\begin{array}{l}\\ 12.& \text{Diketahui jajar genjang ABCD }\\ & \text{dengan titik E adalah perpotongan }\\ & \text{diagonal jajar genjang}.\end{array}$

$\begin{array}{l}\\ .& \text{Jika}\overline{AB}=\overrightarrow{b}\text{dan}\overline{AD}=\overrightarrow{a},\text{maka}\overline{CE}\\ & \text{bila dinyatakan dalam}\overrightarrow{a}\text{dan}\overrightarrow{b}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{2}(\overrightarrow{a}+\overrightarrow{b})}& \\ \text{b}.{\displaystyle \frac{1}{2}(\overrightarrow{a}-\overrightarrow{b})}& \\ \text{c}.{\displaystyle \frac{1}{2}(\overrightarrow{b}-\overrightarrow{a})}& \\ \text{d}.{\displaystyle -\frac{1}{2}(\overrightarrow{a}+\overrightarrow{b})}\\ \text{e}.-{\displaystyle \frac{1}{2}(2\overrightarrow{a}+\overrightarrow{b})}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\overline{AC}& =\overline{AD}+\overline{DC}\\ \overline{CA}& =\overline{CD}+\overline{DA}\\ \overline{CE}& ={\displaystyle \frac{1}{2}\overline{CA}}\\ & ={\displaystyle \frac{1}{2}(-\overrightarrow{b}-\overrightarrow{a})}\\ & =-{\displaystyle \frac{1}{2}(\overrightarrow{a}+\overrightarrow{b})}\end{array}\end{array}$

$\begin{array}{l}\\ 13.& \text{Pada segi enam beraturan ABCDEF},\\ & \text{jika}\overrightarrow{AB}=\overrightarrow{u}\text{dan}\overrightarrow{AF}=\overrightarrow{v}\text{maka vektor}\\ & \overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=....\\ & \begin{array}{l}\\ \text{a}.& 2\overrightarrow{u}+2\overrightarrow{v}& & & \text{d}.& 6\overrightarrow{u}+6\overrightarrow{v}\\ \text{b}.& 4\overrightarrow{u}+4\overrightarrow{v}& \text{c}.& 5\overrightarrow{u}+5\overrightarrow{v}& \text{e}.& 8\overrightarrow{u}+8\overrightarrow{v}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \text{Perhatikanlah ilustrasi gambar berikut}\end{array}$

$\begin{array}{rl}.& \overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}\\ & =\overrightarrow{AB}+(\overrightarrow{AO}+\overrightarrow{OC})+2\overrightarrow{AO}+(\overrightarrow{AO}+\overrightarrow{OE})+\overrightarrow{AF}\\ & =\overrightarrow{u}+(2\overrightarrow{u}+\overrightarrow{v})+2(\overrightarrow{v}+\overrightarrow{u})+(2\overrightarrow{v}+\overrightarrow{u})+\overrightarrow{v}\\ & =6\overrightarrow{u}+6\overrightarrow{v}\end{array}$

$\begin{array}{l}\\ 14.& \text{Perhatikanlah juga ilustrasi gambar berikut}\end{array}$

$\begin{array}{l}\\ .& \text{maka vektor}\overrightarrow{w}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 8\overrightarrow{i}-6\overrightarrow{j}-13\overrightarrow{k}& \\ \text{b}.& 8\overrightarrow{i}-13\overrightarrow{j}-6\overrightarrow{k}\\ \text{c}.& 6\overrightarrow{i}-8\overrightarrow{j}-13\overrightarrow{k}& \\ \text{d}.& -6\overrightarrow{i}+8\overrightarrow{j}-13\overrightarrow{k}\\ \text{e}.& -6\overrightarrow{i}-13\overrightarrow{j}+8\overrightarrow{k}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \text{Kita perhatikan juga ilustrasi }\\ & \text{gambarnya semisal dengan soal No.1}\\ & \text{Misalkan titiknya adalah titik }\\ & \text{W dengan koordinat (8,-6,-13)},\\ & \text{maka vektor posisi titik }\\ & \text{W tersebut adalah}\overrightarrow{OW}=\overrightarrow{w}\\ & \text{di mana}\\ & \begin{array}{rl}& \text{Vektor}\overrightarrow{w}\text{jika dinyatakan }\\ & \text{dalam kombinasi linear adalah}\\ & \overrightarrow{w}=8\overrightarrow{i}-6\overrightarrow{j}-13\overrightarrow{k}\end{array}\end{array}$.

$\begin{array}{l}\\ 15.& \text{Jika titik Z(4,-5,2)},\text{maka panjang }\\ & \text{vektor posisi titik Z adalah}....\\ & \begin{array}{l}\\ \text{a}.& 1& & & \text{d}.& 5\sqrt{2}\\ \text{b}.& 2\sqrt{5}& \text{c}.& 3\sqrt{5}& \text{e}.& 5\sqrt{3}\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\text{Vektor posisi}& \text{titik Z tersebut adalah}\\ \overrightarrow{OZ}=\overrightarrow{z}& =\left(\begin{array}{ccc}4,& -5,& 2\end{array}\right),\\ \text{Dan panjang}& \text{vektor}\overrightarrow{z}\text{ini adalah}\\ \left|\overrightarrow{z}\right|& =\sqrt{4^2+(-5{)}^2+2^2}\\ & =\sqrt{16+25+4}\\ & =\sqrt{45}=\sqrt{9\times 5}\\ & =3\sqrt{5}\end{array}\end{array}$.