To determine the coordinates of point [tex]\( E \)[/tex] that partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex], we follow these steps:
1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (0, 1) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (-1, 3) \)[/tex].
2. Define the ratio components:
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 2 \)[/tex]
3. Substitute the coordinates and ratio components into the formula for the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
x_E = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
Plugging the values in:
[tex]\[
x_E = \left( \frac{1}{1 + 2} \right) ((-1) - 0) + 0
\][/tex]
[tex]\[
x_E = \left( \frac{1}{3} \right) (-1) + 0
\][/tex]
[tex]\[
x_E = -\frac{1}{3} + 0
\][/tex]
[tex]\[
x_E = -\frac{1}{3}
\][/tex]
4. Substitute the coordinates and ratio components into the formula for the [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
y_E = \left( \frac{m}{m+n} \right) (y_2 - y_1) + y_1
\][/tex]
Plugging the values in:
[tex]\[
y_E = \left( \frac{1}{1 + 2} \right) (3 - 1) + 1
\][/tex]
[tex]\[
y_E = \left( \frac{1}{3} \right) (2) + 1
\][/tex]
[tex]\[
y_E = \frac{2}{3} + 1
\][/tex]
[tex]\[
y_E = \frac{2}{3} + \frac{3}{3}
\][/tex]
[tex]\[
y_E = \frac{5}{3}
\][/tex]
Therefore, the coordinates of point [tex]\( E \)[/tex] are:
[tex]\[
\left( -\frac{1}{3}, \frac{5}{3} \right)
\][/tex]
Simplifying these fractions, we get the coordinates of point [tex]\( E \)[/tex] as:
[tex]\[
(-0.3333, 1.6667)
\][/tex]
Thus, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( E \)[/tex], which partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex], are approximately [tex]\( -0.3333 \)[/tex] and [tex]\( 1.6667 \)[/tex], respectively.
