PAS MATEMATIKA BLOG: Lanjutan Contoh Soal Vektor Dimensi Dua (Bagian 3)

$\begin{array}{ll} 7 & Jika \bar{a} = (\begin{array}{c} 4 \\ - 3 \end{array}) dan \bar{b} = (\begin{array}{c} 1 \\ 2 \end{array}) \\ tentukanlah \\ a . \bar{a} + \bar{b} d . (3 \bar{a} + 3 \bar{b}) \\ b . \bar{b} + \bar{a} e . 2 (\bar{a} - \bar{b}) \\ c . 3 (\bar{a} + \bar{b}) f . 2 \bar{a} - 2 \bar{b} \\ Jawab : \\ \begin{aligned} a & \bar{a} + \bar{b} = (\begin{array}{c} 4 \\ - 3 \end{array}) + (\begin{array}{c} 1 \\ 2 \end{array}) = (\begin{array}{c} 4 + 1 \\ - 3 + 2 \end{array}) = (\begin{array}{c} 5 \\ - 1 \end{array}) \\ b & \bar{b} + \bar{a} = (\begin{array}{c} 1 \\ 2 \end{array}) + (\begin{array}{c} 4 \\ - 3 \end{array}) = (\begin{array}{c} 1 + 4 \\ 2 + (- 3) \end{array}) = (\begin{array}{c} 5 \\ - 1 \end{array}) \\ c & \dots \\ d & 3 \bar{a} + 3 \bar{b} = 3 (\begin{array}{c} 4 \\ - 3 \end{array}) + 3 (\begin{array}{c} 1 \\ 2 \end{array}) \\ = (\begin{array}{c} 12 \\ - 9 \end{array}) + (\begin{array}{c} 3 \\ 6 \end{array}) = (\begin{array}{c} 12 + 3 \\ - 9 + 6 \end{array}) = (\begin{array}{c} 15 \\ - 3 \end{array}) \\ e & \dots \\ f & \dots \end{aligned} \end{array}$

$\begin{array}{ll} 8. & Perhatikanlah gambar berikut \end{array}$ .

$. \begin{array}{ll} Pada Balok ABCD.EFGH diatas diketahui \\ DA = 4 cm, DC = 5 cm, dan DH 3 cm . \\ Misalkan \vec{i} adalah vektor satuan dengan arah \\ sama dengan \vec{D A}, \vec{j} adalah vektor satuan \\ dengan arah sama \vec{D C}, dan \vec{k} \\ adalah vektor satuan dengan arah sama dengan \vec{D H} . \\ Nyatakanlah vektor-vektor berikut dalam vektor \vec{i}, \vec{j} dan \vec{k} . \\ \begin{array}{ll} a . & \vec{D A}, \vec{D C} dan \vec{D H} \\ b . & \vec{D B} dan \vec{D F} \\ c . & \vec{D P} jika P titik tengan E F \\ d . & \vec{D Q} jika Q titik pada perpanjangan \\ F G dengan F G = G Q \end{array} \end{array}$ .

$. \begin{aligned} Jawab : \\ \begin{array}{ll} \begin{aligned} a . & {\begin{cases} \vec{D A} = 4 \vec{i} \\ \vec{D C} = 5 \vec{j} \\ \vec{D H} = 3 \vec{k} \end{cases} \end{aligned} & \begin{aligned} b . \vec{D B} & = \vec{D C} + \vec{C B} \\ = \vec{D C} + \vec{D A} \\ = 5 \vec{j} + 4 \vec{i} \\ \vec{D F} & = \vec{D B} + \vec{B F} \\ = 4 \vec{i} + 5 \vec{j} + 3 \vec{k} \end{aligned} \end{array} \\ \begin{array}{ll} \begin{aligned} c . \vec{D P} & = \vec{D E} + \vec{E P} \\ = \vec{D A} + \vec{A E} + \vec{E P} \\ = 4 \vec{i} + 3 \vec{k} + \frac{1}{2} \vec{E F} \\ = 4 \vec{i} + 3 \vec{k} + \frac{1}{2} \vec{D C} \\ = 4 \vec{i} + 3 \vec{k} + \frac{1}{2} (5 \vec{j}) \\ = 4 \vec{i} + \frac{5}{2} \vec{j} + 3 \vec{k} \end{aligned} & \begin{aligned} d . \vec{D Q} & = \vec{D G} + \vec{G Q} \\ = \vec{D C} + \vec{C G} + \vec{G Q} \\ = \vec{D C} + \vec{D H} + \vec{A D} \\ = \vec{D C} + \vec{D H} - \vec{D A} \\ = 5 \vec{j} + 3 \vec{k} - 4 \vec{i} \end{aligned} \end{array} \end{aligned}$

$\begin{array}{ll} 9. & Diketahui \vec{a} = (\begin{array}{c} - 5 \\ 3 \end{array}) dan \vec{b} = (\begin{array}{c} 6 \\ - 2 \end{array}) . \\ Tentukanlah \\ a . \vec{a} - \vec{b} b . \vec{b} - \vec{a} \\ c . 4 (\vec{a} - \vec{b}) d . 4 \vec{a} - 4 \vec{b} \\ Jawab : \\ \begin{array}{llll} \begin{aligned} a . \vec{a} - \vec{b} & = \vec{a} + (- \vec{b}) \\ = (\begin{array}{c} - 5 \\ 3 \end{array}) + (\begin{array}{c} - 6 \\ 2 \end{array}) \\ = (\begin{array}{c} - 11 \\ 5 \end{array}) \end{aligned} \\ \begin{aligned} b . \vec{b} - \vec{a} & = \vec{b} + (- \vec{a}) \\ = (\begin{array}{c} 6 \\ - 2 \end{array}) + (\begin{array}{c} 5 \\ - 3 \end{array}) \\ = (\begin{array}{c} 11 \\ - 5 \end{array}) \end{aligned} \\ \begin{aligned} c . 4 (\vec{a} - \vec{b}) & = 4 (\begin{array}{c} - 11 \\ 5 \end{array}) \\ = (\begin{array}{c} - 44 \\ 20 \end{array}) \end{aligned} \end{array} \end{array}$

$\begin{array}{ll} 10. & Diketahui \vec{p} = (\begin{array}{c} 4 \\ - 3 \end{array}) dan \vec{q} = (\begin{array}{c} 1 \\ 2 \end{array}) . \\ Tentukanlah \\ a . \vec{p} + \vec{q} b . \vec{q} + \vec{p} \\ c . 4 (\vec{p} + \vec{q}) d . 4 \vec{p} + 4 \vec{q} \\ e . 4 (\vec{p} - \vec{q}) f . 4 \vec{p} - 4 \vec{q} \\ Jawab : \\ \begin{array}{llll} \begin{aligned} a . \vec{p} + \vec{q} & = (\begin{array}{c} 4 \\ - 3 \end{array}) + (\begin{array}{c} 1 \\ 2 \end{array}) \\ = (\begin{array}{c} 5 \\ - 1 \end{array}) \end{aligned} & \begin{aligned} b . \vec{q} + \vec{p} & = (\begin{array}{c} 1 \\ 2 \end{array}) + (\begin{array}{c} 4 \\ - 3 \end{array}) \\ = (\begin{array}{c} 5 \\ - 1 \end{array}) \end{aligned} \end{array} \end{array}$

$\begin{array}{ll} 11. & Pada contoh soal No. 8 tentukanlah \\ panjang vektor \vec{D A}, \vec{D P} dan \vec{D Q} \\ Jawab : \\ \begin{aligned} | \vec{D A} | = 4 c m, \\ {| \vec{D P} |}^{2} = {| \vec{D E} |}^{2} + {| \vec{E P} |}^{2} \\ = {| \vec{D A} |}^{2} + {| \vec{A E} |}^{2} + {| \vec{E P} |}^{2} \\ = 4^{2} + 3^{2} + {(\frac{5}{2})}^{2} \\ = 16 + 9 + \frac{25}{4} \\ | \vec{D P} | = \sqrt{\frac{125}{4}} \\ = \frac{5}{2} \sqrt{5} c m, dan \\ {| \vec{D Q} |}^{2} = {| \vec{D G} |}^{2} + {| \vec{G Q} |}^{2} \\ = {| \vec{D C} |}^{2} + {| \vec{C G} |}^{2} + {| \vec{G Q} |}^{2} \\ = 5^{2} + 3^{2} + 4^{2} \\ = 25 + 9 + 16 \\ = 50 \\ | \vec{D Q} | & = \sqrt{50} \\ = 5 \sqrt{2} c m \end{aligned} \end{array}$

$\begin{array}{ll} 12. & Perhatikanlah gambar di bawah ini \\ dan nyatakan titik-titik pada gambar \\ tersebut dalam vektor posisi \end{array}$

. \begin{aligned} Jawab : \\ Perhatikanlah gambar berikut \end{aligned}

. \begin{array}{ll} Titik & Vektor Posisi \\ P (5, 3) & \vec{p} = (\begin{array}{c} 5 \\ 3 \end{array}) \\ Q (2, - 3) & \vec{q} = (\begin{array}{c} 2 \\ - 3 \end{array}) \\ R (- 5, - 1) & \vec{r} = (\begin{array}{c} - 5 \\ - 1 \end{array}) \\ S (- 3, 7) & \vec{s} = (\begin{array}{c} - 3 \\ 7 \end{array}) \end{array}

\begin{array}{ll} 13. & Perhatikanlah gambar pada soal No. 12 di atas. \\ Tentukanlah vektor-vektor berikut \\ a . \vec{P Q} \\ b . \vec{P S} \\ c . \vec{Q S} \\ d . \vec{Q P} + \vec{P S} \\ Jawab : \\ \begin{array}{ll} \begin{aligned} a . \vec{P Q} & = \vec{P O} + \vec{O Q} \\ = - \vec{O P} + \vec{O Q} \\ = \vec{O Q} - \vec{O P} \\ = \vec{q} - \vec{p} \\ = (\begin{array}{c} 2 \\ - 3 \end{array}) - (\begin{array}{c} 5 \\ 3 \end{array}) \\ = (\begin{array}{c} - 3 \\ - 6 \end{array}) \end{aligned} & \begin{aligned} b . \vec{P S} & = \vec{s} - \vec{p} \\ dengan & cara semisal poin a \\ \vec{P S} & = (\begin{array}{c} - 3 \\ 7 \end{array}) - (\begin{array}{c} 5 \\ 3 \end{array}) \\ = (\begin{array}{c} - 8 \\ 4 \end{array}) \end{aligned} \end{array} \\ \begin{array}{ll} \begin{aligned} c . \vec{Q S} & = \vec{s} - \vec{q} \\ = (\begin{array}{c} - 3 \\ 7 \end{array}) - (\begin{array}{c} 2 \\ - 3 \end{array}) \\ = (\begin{array}{c} - 5 \\ 10 \end{array}) \end{aligned} & \begin{aligned} d . \vec{Q P} + \vec{P S} & = \vec{Q S} \\ = (\begin{array}{c} - 5 \\ 10 \end{array}) \\ lihat jawaban & poin c \end{aligned} \end{array} \end{array}

PAS MATEMATIKA BLOG

Lanjutan Contoh Soal Vektor Dimensi Dua (Bagian 3)

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