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] of the length of the segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula for internal division.
Given points [tex]\( A(x_1, y_1) = (2, -1) \)[/tex] and [tex]\( B(x_2, y_2) = (4, -3) \)[/tex], let's calculate the coordinates step-by-step:
1. We know that [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( \frac{2}{3} \)[/tex]. Therefore, we can denote [tex]\( m = 2 \)[/tex] and [tex]\( n = 1 \)[/tex]. This comes from the ratio [tex]\( 2 \)[/tex] parts of [tex]\( AB \)[/tex] to [tex]\( 1 \)[/tex] part of [tex]\( AP \)[/tex] added together giving [tex]\( m + n = 3 \)[/tex].
2. Substitute the ratio and coordinates into the section formula:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
Plugging in the values:
[tex]\[
x = \left(\frac{2}{3}\right)(4 - 2) + 2
\][/tex]
Now calculate the [tex]\( x \)[/tex]-coordinate:
[tex]\[
x = \left(\frac{2}{3}\right)(2) + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33
\][/tex]
Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \left(\frac{2}{3}\right)(-3 - (-1)) + (-1)
\][/tex]
[tex]\[
y = \left(\frac{2}{3}\right)(-2) - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[
\left( 3.33, -2.33 \right)
\][/tex]
