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tolong jawabDiketahui P pada AB, tentukan lah kordinat P jika : a. A (-2 , 3), B (3, 7), dan AP :

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Diketahui P pada AB, tentukanlah koordinat P seandainya :

a. A(-2, 3), B(3, 7), dan AP : PB = 3 : 2

b. A(-3, -2, -1), B(0, -5, 2), dan AP : PB = 4 : 3

c. A(5, 2, 1), B(9, 10, 13), dan AP : PB= 4 : 1

(Jawaban:

a.

$P(1,\frac{27}{5})$

b.
$P(-\frac{9}{7},-\frac{26}{7},\frac{5}{7})$

c.

$P(\frac{41}{5},\frac{42}{5},\frac{53}{5})$
)

Cak bagi menyelesaikan permasalahan di atas, Anda dapat menerapkan konsep
vektor
yaitu

$\vec{AB}=B-A$

Penjelasan dengan awalan-langkah:

Diketahui:

a. A(-2, 3), B(3, 7), dan AP : PB = 3 : 2

b. A(-3, -2, -1), B(0, -5, 2), dan AP : PB = 4 : 3

c. A(5, 2, 1), B(9, 10, 13), dan AP : PB= 4 : 1

Ditanya:

Tentukanlah koordinat P lega AB.

Jawab:

a. Kita n kepunyaan

$\frac{\vec{AP}}{\vec{PB}}=\frac{3}{2}$

$2\vec{AP}=3\vec{PB}$

$2(P-A)=3(B-P)$

$2P-2A=3B-3P$

$5P=2A+3B$

$P=\frac{2}{5} A+\frac{3}{5} B$

$P=\frac{2}{5} \left[\begin{array}{ccc}-2\\3\end{array}\right]+\frac{3}{5} \left[\begin{array}{ccc}3\\7\end{array}\right]$

$P=\left[\begin{array}{ccc}-\frac{4}{5} \\\frac{6}{5} \end{array}\right]+ \left[\begin{array}{ccc}\frac{9}{5} \\\frac{21}{5} \end{array}\right]$

$P=\left[\begin{array}{ccc}1 \\\frac{27}{5} \end{array}\right]$

b. Kita memiliki

$\frac{\vec{AP}}{\vec{PB}}=\frac{4}{3}$

$3\vec{AP}=4\vec{PB}$

$3(P-A)=4(B-P)$

$3P-3A=4B-4P$

$7P=3A+4B$

$P=\frac{3}{7} A+\frac{4}{7} B$

$P=\frac{3}{7} \left[\begin{array}{ccc}-3\\-2\\-1\end{array}\right] +\frac{4}{7}\left[\begin{array}{ccc}0\\-5\\2\end{array}\right]$

$P=\left[\begin{array}{ccc}-\frac{9}{7} \\-\frac{6}{7} \\-\frac{3}{7} \end{array}\right] +\left[\begin{array}{ccc}0\\-\frac{20}{7} \\\frac{8}{7} \end{array}\right]$

$P=\left[\begin{array}{ccc}-\frac{9}{7} \\-\frac{26}{7} \\\frac{5}{7} \end{array}\right]$

c. Kita memiliki

$\frac{\vec{AP}}{\vec{PB}}=\frac{4}{1}$

$\vec{AP}=4\vec{PB}$

$P-A=4(B-P)$

$P-A=4B-4P$

$5P=A+4B$

$P=\frac{1}{5} A+\frac{4}{5} B$

$P=\frac{1}{5} \left[\begin{array}{ccc}5\\2\\1\end{array}\right] +\frac{4}{5} \left[\begin{array}{ccc}9\\10\\13\end{array}\right]$

$P=\left[\begin{array}{ccc}1\\\frac{2}{5} \\\frac{1}{5} \end{array}\right] +\left[\begin{array}{ccc}\frac{36}{5} \\8\\\frac{52}{5} \end{array}\right]$

$P=\left[\begin{array}{ccc}\frac{41}{5} \\\frac{42}{5} \\\frac{53}{5} \end{array}\right]$

Makara, diperoleh koordinat titik P yaitu

a.
$P(1,\frac{27}{5})$

b.
$P(-\frac{9}{7},-\frac{26}{7},\frac{5}{7})$

c.
$P(\frac{41}{5},\frac{42}{5},\frac{53}{5})$

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