To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{2}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we will use the section formula. Given coordinates:
- Point [tex]\( A \)[/tex] has coordinates [tex]\((2, -1)\)[/tex]
- Point [tex]\( B \)[/tex] has coordinates [tex]\((4, -3)\)[/tex]
The ratio [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( \frac{2}{3} \)[/tex]. We interpret this as [tex]\( P \)[/tex] dividing [tex]\( AB \)[/tex] in the ratio [tex]\( 2:1 \)[/tex] because it is [tex]\(\frac{2}{3}\)[/tex] of the way, meaning [tex]\( \frac{2}{3} \cdot 3 = 2\)[/tex].
### Using the Section Formula
The section formula states:
[tex]\[
\begin{aligned}
x_P & = \frac{m \cdot x2 + n \cdot x1}{m + n}, \\
y_P & = \frac{m \cdot y2 + n \cdot y1}{m + n},
\end{aligned}
\][/tex]
where [tex]\( A = (x1, y1) \)[/tex], [tex]\( B = (x2, y2) \)[/tex], and [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the ratios.
In our case:
- [tex]\( x1 = 2 \)[/tex],
- [tex]\( y1 = -1 \)[/tex],
- [tex]\( x2 = 4 \)[/tex],
- [tex]\( y2 = -3 \)[/tex],
- [tex]\( m = 2 \)[/tex],
- [tex]\( n = 1 \)[/tex].
Substituting these values into the formula, we find:
[tex]\[
x_P = \left(\frac{2}{2+1}\right) \left(x2 - x1\right) + x1
\][/tex]
[tex]\[
y_P = \left(\frac{2}{2+1}\right) \left(y2 - y1\right) + y1
\][/tex]
Substituting the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
x_P = \left(\frac{2}{2 + 1}\right) \left(4 - 2\right) + 2 = \left(\frac{2}{3}\right) \cdot 2 + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33
\][/tex]
[tex]\[
y_P = \left(\frac{2}{2 + 1}\right) \left(-3 - (-1)\right) + (-1) = \left(\frac{2}{3}\right) \cdot (-2) + (-1) = \left(\frac{2}{3}\right) \cdot -2 - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are approximately:
[tex]\[
(3.333333333333333, -2.333333333333333)
\][/tex]
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (3, -2) \)[/tex].
