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The endpoints of the directed line segment AB are A(−7, 4) and B(8, 9). Find the coordinates of point P along AB¯¯¯¯¯¯¯¯ so that the ratio of AP to PB is 2 to 3.

Answer :

xero099

Answer:

The coordinates of point P along the segment AB is (-1, 6).

Step-by-step explanation:

It is known that ratio of AP to PB is:

[tex]\frac{AP}{PB} = \frac{2}{3}[/tex]

Or vectorially speaking:

[tex]\overrightarrow{AP} = \frac{2}{3}\overrightarrow{PB}[/tex]

The vector that represents the segment AB is:

[tex]\overrightarrow{AB} = \vec B -\vec A[/tex]

If [tex]\vec A = (-7,4)[/tex] and [tex]\vec B = (8,9)[/tex], then:

[tex]\overrightarrow{AB} = (8,9)-(-7,4)[/tex]

[tex]\overrightarrow {AB} = (8+7,9-4)[/tex]

[tex]\overrightarrow{AB} = (15,5)[/tex]

But [tex]\overrightarrow{AB} = \overrightarrow{AP} + \overrightarrow{PB}[/tex], then:

[tex]\overrightarrow{AB} = \frac{2}{3}\overrightarrow{PB} +\overrightarrow{PB}[/tex]

[tex]\overrightarrow{AB} = \frac{5}{3}\overrightarrow{PB}[/tex]

[tex]\overrightarrow{PB} = \frac{3}{5}\overrightarrow{AB}[/tex]

[tex]\overrightarrow{PB} = \frac{3}{5}(15,5)[/tex]

[tex]\overrightarrow{PB} = \left(9, 3\right)[/tex]

The location of [tex]\vec P[/tex] is derived of this formula:

[tex]\overrightarrow{PB} = \vec B - \vec P[/tex]

[tex]\vec P = \vec B - \overrightarrow{PB}[/tex]

If [tex]\vec B = (8,9)[/tex] and [tex]\overrightarrow{PB} = \left(9, 3\right)[/tex], then:

[tex]\vec P = (8,9) - (9,3)[/tex]

[tex]\vec P = (8-9,9-3)[/tex]

[tex]\vec P = (-1, 6)[/tex]

The coordinates of point P along the segment AB is (-1, 6).

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