Contoh Soal Vektor di Dimensi Dua (Matematika Peminatan Kelas X) Bagian 1

 Perhatikanlah gambar berikut untuk menjawab soal no.1

$\begin{array}{l}\\ 1.& \text{Jika}\overline{XW}=\mathbf{\text{a}},\overline{XY}=\mathbf{\text{b}},\text{dan}\overline{YZ}=\mathbf{\text{c}}\\ & \text{Nyatakan dalam vektor},\mathbf{\text{a}},\mathbf{\text{b}},\mathrm{\&}\mathbf{\text{c}}\\ & \text{untuk vektor-vektor berikut}\\ & \text{a}.\overline{WY}\text{d}.\overline{WZ}\\ & \text{b}.\overline{XZ}\text{e}.\overline{WM}\\ & \text{c}.\overline{ZX}\text{f}.\overline{MY}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{a}.\overline{WY}& =\overline{WX}+\overline{XY}=\mathbf{\text{a}}+\mathbf{\text{b}}\\ \text{b}.\overline{XZ}& =\overline{XY}+\overline{YZ}=\mathbf{\text{b}}+\mathbf{\text{c}}\\ \text{c}.\overline{ZX}& =\overline{ZY}+\overline{YX}=-\mathbf{\text{c}}+(-\mathbf{\text{b}})=-\mathbf{\text{b}}-\mathbf{\text{c}}\\ & =-(\mathbf{\text{b}}+\mathbf{\text{c}})\\ & \text{atau}\\ \overline{ZX}& =-\overline{XZ}=-(\mathbf{\text{b}}+\mathbf{\text{c}})\\ \text{d}.\overline{WZ}& =\overline{WX}+\overline{XY}+\overline{YZ}\\ & =\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}}\\ \text{e}.\overline{WM}& ={\displaystyle \frac{1}{2}\overline{WZ}}\\ & ={\displaystyle \frac{1}{2}(\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}})}\\ \text{f}.\overline{MY}& =\overline{MZ}+\overline{ZY}\\ & ={\displaystyle \frac{1}{2}\overline{WZ}+\overline{ZY}}\\ & ={\displaystyle \frac{1}{2}(\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}})+(-\mathbf{\text{c}})}\\ & ={\displaystyle \frac{1}{2}(\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}})-\mathbf{\text{c}}}\\ & ={\displaystyle \frac{1}{2}(\mathbf{\text{a}}+\mathbf{\text{b}}-\mathbf{\text{c}})}\end{array}\end{array}$

Perhatikanlah gambar berikut untuk menjawab soal no.2

$\begin{array}{l}\\ 2.& \text{Jika}\overline{PQ}=\mathbf{\text{a}},\overline{QR}=\mathbf{\text{b}},\text{dan}\overline{RS}=\mathbf{\text{c}}\\ & \text{dan titik}\mathbf{\text{E}}\text{dan}\mathbf{\text{F}}\text{adalah titik tegah}\\ & \overline{RS}\text{dan}\overline{QS},\\ & \text{nyatakanlah dalam vektor},\mathbf{\text{a}},\mathbf{\text{b}},\mathrm{\&}\mathbf{\text{c}}\\ & \text{untuk vektor-vektor berikut}\\ & \text{a}.\overline{PR}\text{e}.\overline{PF}\\ & \text{b}.\overline{RP}\text{f}.\overline{SF}\\ & \text{c}.\overline{PS}\text{g}.\overline{FR}\\ & \text{d}.\overline{QE}\text{h}.\overline{EF}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{a}.\overline{PR}& =\overline{PQ}+\overline{QR}=\mathbf{\text{a}}+\mathbf{\text{b}}\\ \text{b}.\overline{RP}& =\overline{RQ}+\overline{QP}=-\mathbf{\text{b}}-\mathbf{\text{a}}=-(\mathbf{\text{a}}+\mathbf{\text{b}})\\ \text{c}.\overline{PS}& =\overline{PQ}+\overline{QR}+\overline{RS}=\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}}\\ \text{d}.\overline{QE}& =\overline{QR}+\overline{RE}=\overline{QR}+{\displaystyle \frac{1}{2}\overline{RS}}\\ & =\mathbf{\text{b}}+{\displaystyle \frac{1}{2}\mathbf{\text{c}}}\\ \text{e}.\overline{PF}& =\overline{PQ}+\overline{QF}=\overline{PQ}+{\displaystyle \frac{1}{2}\overline{QS}}\\ & =\overline{PQ}+{\displaystyle \frac{1}{2}(\overline{QR}+\overline{RS})}\\ & ={\displaystyle \frac{1}{2}(2\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}})}\\ \text{f}.\overline{SF}& ={\displaystyle \frac{1}{2}\overline{SQ}={\displaystyle \frac{1}{2}(\overline{SR}+\overline{RQ})={\displaystyle \frac{1}{2}(-\mathbf{\text{c}}+(-\mathbf{\text{b}}))}}}\\ & =-{\displaystyle \frac{1}{2}(\mathbf{\text{b}}+\mathbf{\text{c}})}\\ \text{g}.\overline{FR}& =\overline{FQ}+\overline{QR}={\displaystyle \frac{1}{2}\overline{SQ}+\overline{QR}}\\ & =-{\displaystyle \frac{1}{2}(\mathbf{\text{b}}+\mathbf{\text{c}})+\mathbf{\text{b}}={\displaystyle \frac{1}{2}(\mathbf{\text{b}}-\mathbf{\text{c}})}}\\ \text{h}.\overline{EF}& =\overline{ES}+\overline{SF}={\displaystyle \frac{1}{2}\overline{RS}+{\displaystyle \frac{1}{2}\overline{SQ}}}\\ & ={\displaystyle \frac{1}{2}\mathbf{\text{c}}+(-{\displaystyle \frac{1}{2}(\mathbf{\text{b}}+\mathbf{\text{c}})})=-{\displaystyle \frac{1}{2}\mathbf{\text{b}}}}\end{array}\end{array}$

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

$\begin{array}{l}\\ .& \text{Jika pada titik P bekerja 3 buah gaya}\\ & \text{seperti pada gambar di bawah, lukislah}\\ & \text{vektor}\\ & \mathbf{\text{r}}=\mathbf{\text{a}}+\mathbf{\text{b}}+\mathbf{\text{c}}\\ \\ & \text{Jawab}:\\ & \text{Dengan aturan poligon kita akan}\\ & \text{mendapatkan gambar berikut}\end{array}$