Contoh Soal Vektor di Dimensi Dua (Kelas X Matematika Peminatan) Bagian 2

 $\begin{array}{l}\\ 4.& \text{Perhatikanlah ilustrasi gambar berikut}\end{array}$

$\begin{array}{l}\\ .& \text{Nyatakanlah vektor-vektor di atas dalam}\\ & \text{a}.\text{vektor kolom}\\ & \text{b}.\text{vektor baris}\\ & \text{c}.\text{vektor basis}\\ \\ & \text{Jawab}:\\ & \text{Vektor di atas saat dinyatakan dengan}\\ & \begin{array}{|clll|}\hline \text{No}& \text{Kolom}& \text{Baris}& \text{Basis}\\ 1& \overline{a}=\left(\begin{array}{c}2\\ 4\end{array}\right)& \overline{a}=\left(\begin{array}{cc}2,& 4\end{array}\right)& \overline{a}=2\overline{i}+4\overline{j}\\ 2& \overline{b}=\left(\begin{array}{c}4\\ 2\end{array}\right)& \overline{b}=\left(\begin{array}{cc}4,& 2\end{array}\right)& \overline{b}=4\overline{i}+2\overline{j}\\ 3& \overline{c}=\left(\begin{array}{c}5\\ 0\end{array}\right)& \overline{c}=\left(\begin{array}{cc}5,& 0\end{array}\right)& \overline{c}=5\overline{i}\\ 4& \overline{d}=\left(\begin{array}{c}-2\\ -4\end{array}\right)& \overline{d}=\left(\begin{array}{cc}-2,& -4\end{array}\right)& \overline{d}=-2\overline{i}-4\overline{j}\\ 5& \overline{e}=\left(\begin{array}{c}0\\ -4\end{array}\right)& \overline{e}=\left(\begin{array}{cc}0,& -4\end{array}\right)& \overline{e}=-4\overline{j}\\ 6& \overline{f}=\left(\begin{array}{c}3\\ -3\end{array}\right)& \overline{f}=\left(\begin{array}{cc}3,& -3\end{array}\right)& \overline{f}=3\overline{i}-3\overline{j}\\ \hline\end{array}\end{array}$

$\begin{array}{l}\\ 5& \text{Tentukanlah panjang atau besar dari}\\ & \text{vektor-vektor berikut}\\ & \text{a}.\overline{a}=\left(\begin{array}{c}-4\\ 3\end{array}\right)\text{d}.\overline{d}=\left(\begin{array}{cc}6,& -8\end{array}\right)\\ & \text{b}.\overline{b}=\left(\begin{array}{c}5\\ 0\end{array}\right)\text{e}.\overline{e}=2\overline{i}+4\overline{j}\\ & \text{c}.\overline{c}=\left(\begin{array}{cc}4,& -6\end{array}\right)\text{f}.\overline{f}=-5\overline{i}+12\overline{j}\\ \\ & \text{Jawab}:\\ & \text{Lambang panjang suatu vektor adalah}:\\ & \left|\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)\right|=\sqrt{u_1^2+u_2^2},\text{maka}\\ & \bullet \left|\overline{a}\right|=\sqrt{(-4{)}^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\\ & \bullet \left|\overline{b}\right|=\sqrt{5^2+0^2}=\sqrt{25}=5\\ & \bullet \left|\overline{c}\right|=\sqrt{4^2+6^2}=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}\\ & \bullet \left|\overline{d}\right|=\sqrt{6^2+(-8{)}^2}=\sqrt{36+64}=\sqrt{100}=10\\ & \bullet \left|\overline{e}\right|=\sqrt{2^2+4^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}\\ & \bullet \left|\overline{f}\right|=\sqrt{(-5{)}^2+{12}^2}=\sqrt{25+144}=\sqrt{169}=13\end{array}$

$\begin{array}{l}\\ 6& \text{Pada soal no.4 di atas dengan menggunakan}\\ & \text{aturan segitiga dan jajar genjang, gambarlah}\\ & \text{vektor-vektor berikut pada kertas berpertak}\\ & \text{a}.\overline{a}+\overline{b}\text{d}.(\overline{a}+\overline{b})+\overline{c}\\ & \text{b}.\overline{b}+\overline{c}\text{e}.2\overline{a}+\overline{e}+2\overline{b}+3\overline{d}\\ & \text{c}.\overline{a}+(\overline{b}+\overline{c})\text{f}.2(\overline{a}+\overline{b}+\overline{c}+\overline{d}+\overline{e})\\ \\ & \text{Jawab}:\end{array}$