Question

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 1

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:

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

Or vectorially speaking:

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

The vector that represents the segment AB is:

$\overrightarrow{AB} = \vec B -\vec A$

If $\vec A = (-7,4)$ and $\vec B = (8,9)$, then:

$\overrightarrow{AB} = (8,9)-(-7,4)$

$\overrightarrow {AB} = (8+7,9-4)$

$\overrightarrow{AB} = (15,5)$

But $\overrightarrow{AB} = \overrightarrow{AP} + \overrightarrow{PB}$, then:

$\overrightarrow{AB} = \frac{2}{3}\overrightarrow{PB} +\overrightarrow{PB}$

$\overrightarrow{AB} = \frac{5}{3}\overrightarrow{PB}$

$\overrightarrow{PB} = \frac{3}{5}\overrightarrow{AB}$

$\overrightarrow{PB} = \frac{3}{5}(15,5)$

$\overrightarrow{PB} = \left(9, 3\right)$

The location of $\vec P$ is derived of this formula:

$\overrightarrow{PB} = \vec B - \vec P$

$\vec P = \vec B - \overrightarrow{PB}$

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

$\vec P = (8,9) - (9,3)$

$\vec P = (8-9,9-3)$

$\vec P = (-1, 6)$

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