Certainly! Let's find the coordinates of point [tex]\( P \)[/tex] that partitions the directed line segment [tex]\( \overline{AB} \)[/tex] from [tex]\( A(-2, 0) \)[/tex] to [tex]\( B(8, 5) \)[/tex] in the ratio [tex]\( 3:2 \)[/tex].
To determine the coordinates of point [tex]\( P \)[/tex], we use the section formula for a point dividing a line segment internally in a given ratio. The formula for a point [tex]\( P(x, y) \)[/tex] that divides the line segment joining two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
P_x = \frac{mx_2 + nx_1}{m + n}
\][/tex]
[tex]\[
P_y = \frac{my_2 + ny_1}{m + n}
\][/tex]
Here, the coordinates of [tex]\( A \)[/tex] are [tex]\( (-2, 0) \)[/tex], the coordinates of [tex]\( B \)[/tex] are [tex]\( (8, 5) \)[/tex], and the ratio [tex]\( m:n \)[/tex] is [tex]\( 3:2 \)[/tex].
Let's substitute these values into the section formula.
1. Calculate [tex]\( P_x \)[/tex]:
[tex]\[
P_x = \frac{3 \cdot 8 + 2 \cdot (-2)}{3 + 2}
\][/tex]
[tex]\[
P_x = \frac{24 - 4}{5}
\][/tex]
[tex]\[
P_x = \frac{20}{5}
\][/tex]
[tex]\[
P_x = 4
\][/tex]
2. Calculate [tex]\( P_y \)[/tex]:
[tex]\[
P_y = \frac{3 \cdot 5 + 2 \cdot 0}{3 + 2}
\][/tex]
[tex]\[
P_y = \frac{15 + 0}{5}
\][/tex]
[tex]\[
P_y = \frac{15}{5}
\][/tex]
[tex]\[
P_y = 3
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] that partitions the line segment [tex]\( \overline{AB} \)[/tex] in the ratio [tex]\( 3:2 \)[/tex] are [tex]\( (4, 3) \)[/tex].
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \boxed{(4, 3)} \)[/tex].
