Contoh 7 Soal dan Pembahasan Materi Vektor

$\begin{array}{l}\\ 31.& \text{Diketahui}\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=1,\text{dan}|\overrightarrow{a}-\overrightarrow{b}|=1\\ & \text{maka panjang vektor}\overrightarrow{a}+\overrightarrow{b}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& \sqrt{3}& & & \text{d}.& 2\sqrt{2}\\ \text{b}.& \sqrt{5}& \text{c}.& \sqrt{7}& \text{e}.& 3\end{array}\\ \\ & \mathbf{\text{Jawab}}\\ & \begin{array}{rl}\text{Diketahui}& \text{sebagaimana pada soal}\\ {|\overrightarrow{a}-\overrightarrow{b}|}^2& ={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2-2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ 1^2& ={\left(\sqrt{3}\right)}^2+1^2-2.\sqrt{3}.1.\cos \theta \\ 2\sqrt{3}\cos \theta & =3\\ \text{maka pan}& \text{jang vektor}\overrightarrow{a}+\overrightarrow{b}\text{adalah}\\ |\overrightarrow{a}+\overrightarrow{b}|& =\sqrt{{\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta }\\ & =\sqrt{{\left(\sqrt{3}\right)}^2+1^2+3}\\ & =\sqrt{3+1+3}\\ & =\sqrt{7}\end{array}\end{array}$

$\begin{array}{l}\\ 32.& \text{Jika}\left|\bar{u}\right|=6,\left|\bar{v}\right|=4\sqrt{3},\text{dan}|\bar{u}-\bar{v}|=8\\ & \text{tentukanlah nilai dari}\\ & \text{a}.\bar{u}\bullet \bar{v}\\ & \text{b}.|\bar{u}+\bar{v}|\\ & \text{c}.\mathbf{\text{cosinus}}\text{sudut antara}\bar{u}\text{dan}\bar{v}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{a}.& \bar{u}\bullet \bar{v}=\cdots \\ & 2.\bar{u}\bullet \bar{v}={\left|\bar{u}\right|}^2+{\left|\bar{v}\right|}^2-{|\bar{u}-\bar{v}|}^2\\ & 2.\bar{u}\bullet \bar{v}=6^2+(4\sqrt{3}{)}^2-8^2\\ & 2.\bar{u}\bullet \bar{v}=36+48-64=84-64=20\\ & \bar{u}\bullet \bar{v}={\displaystyle \frac{20}{2}=10}\\ \text{b}.& {|\bar{u}+\bar{v}|}^2={\left|\bar{u}\right|}^2+{\left|\bar{v}\right|}^2+2.\bar{u}\bullet \bar{v}\\ & {|\bar{u}+\bar{v}|}^2=6^2+(4\sqrt{3}{)}^2+20\\ & =84+20=104\\ & |\bar{u}+\bar{v}|=\sqrt{104}\\ \text{c}.& \cos \mathrm{\angle }(\bar{u},\bar{v})={\displaystyle \frac{\bar{u}\bullet \bar{v}}{\left|\bar{u}\right|.\left|\bar{v}\right|}=\frac{10}{6.(4\sqrt{3})}\times \frac{\sqrt{3}}{\sqrt{3}}}\\ & ={\displaystyle \frac{10\sqrt{3}}{72}=\frac{5}{36}\sqrt{3}}\\ & \mathbf{\text{Berikut ilustrasi gambarnya}}\end{array}\end{array}$.

$\begin{array}{l}\\ 33.& \text{Jika}\overrightarrow{p}=\left(\begin{array}{c}2\\ -5\end{array}\right),\overrightarrow{q}=\left(\begin{array}{c}4\\ 3\end{array}\right),\text{maka }\\ & \text{proyeksi skalar ortogonal vektor}\overrightarrow{p}\\ & \text{pada}\overrightarrow{q}\text{adalah}....\\ & \text{a}.{\displaystyle \frac{3}{5}}\\ \\ & \text{b}.{\displaystyle \frac{7}{5}}\\ \\ & \text{c}.{\displaystyle \frac{8}{5}}\\ \\ & \text{d}.{\displaystyle \frac{9}{5}}\\ \\ & \text{e}.{\displaystyle 2}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\left|\overrightarrow{r}\right|& ={\displaystyle \frac{\overrightarrow{p}.\overrightarrow{q}}{\left|\overrightarrow{q}\right|}}\\ & ={\displaystyle \frac{\left(\begin{array}{c}2\\ -5\end{array}\right).\left(\begin{array}{c}4\\ 3\end{array}\right)}{\sqrt{4^2+3^2}}}\\ & ={\displaystyle \frac{8+(-15)}{\sqrt{25}}}\\ & =\left|{\displaystyle \frac{-7}{5}}\right|\\ & ={\displaystyle \frac{7}{5}}\end{array}\end{array}$

$\begin{array}{l}\\ 34.& \text{Panjang Proyeksi vektor}\overrightarrow{a}=\left(\begin{array}{c}5\\ 1\end{array}\right)\\ & \text{pada}\overrightarrow{b}=\left(\begin{array}{c}0\\ 4\end{array}\right)\text{adalah}....\\ & \text{a}.{\displaystyle -1}\\ & \text{b}.-{\displaystyle \frac{1}{2}}\\ & \text{c}.1\\ & \text{d}.{\displaystyle 2}\\ & \text{e}.4\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\left|\overrightarrow{c}\right|& =\left|{\displaystyle \frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right|\\ & =\left|{\displaystyle \frac{\left(\begin{array}{c}5\\ 1\end{array}\right)\bullet \left(\begin{array}{c}0\\ -4\end{array}\right)}{\left|\sqrt{0^2+(-4{)}^2}\right|}}\right|\\ & =\left|{\displaystyle \frac{0-4}{4}}\right|=|-1|=1\end{array}\end{array}$

$\begin{array}{l}\\ 35.& \text{Proyeksi vektor ortogonal}\overrightarrow{a}=\left(\begin{array}{c}2\\ -4\end{array}\right)\\ & \text{pada}\overrightarrow{b}=\left(\begin{array}{c}-1\\ 2\end{array}\right)\text{adalah}....\\ & \text{a}.\left(\begin{array}{c}2\\ -1\end{array}\right)\\ & \text{b}.\left(\begin{array}{c}2\\ -2\end{array}\right)\\ & \text{c}.\left(\begin{array}{c}2\\ -4\end{array}\right)\\ & \text{d}.\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & \text{e}.\left(\begin{array}{c}-2\\ 4\end{array}\right)\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overrightarrow{c}& =\left({\displaystyle \frac{\overrightarrow{a}\bullet \overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}}\right).\overrightarrow{b}\\ & =\left({\displaystyle \frac{\left(\begin{array}{c}2\\ -4\end{array}\right)\bullet \left(\begin{array}{c}-1\\ 2\end{array}\right)}{(-1{)}^2+2^2}}\right).\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =\left({\displaystyle \frac{-2-8}{1+4}}\right).\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =-2\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =\left(\begin{array}{c}2\\ -4\end{array}\right)\end{array}\end{array}$

DAFTAR PUSTAKA

1. Johanes, Kastolan, Sulasim. 2006. Kompetensi Matematika Program IPA 3A SMA Kelas XII Semester Pertama. Jakarta: YUDHISTIRA.
2. Kusnandar, Muharman, I., Indrianti, M. 2017. Pendalaman Buku Teks Matematika SMA Kelas X Peminatan MIPA. Jakarta: YUDHISTIRA.
3. Miyanto, Aksin, N., Suparno. 2021. Buku Interaktif Matematika untuk SMA/MA Peminatan Matematika dan Ilmu-Ilmu Alam Kelas X Semester 2. Yogyakarta: INTAN PARIWARA. 
4. Sembiring, S., Zulkifli, M., Marsito, Rusdi, I. 2016. Matematika untuk Siswa SMA/MA Kelas X Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Bandung: SRIKANDI EMPAT WIDYA UTAMA.
5. Yuana, R.A., Indriyastuti. 2017. Persektif Matematika untuk Kelas X SMA dan MA Kelompok Peminatan Matematika dan Ilmu Alam. Solo: PT TIGA SERANGKAI MANDIRI.