To find the coordinates of point [tex]\( E \)[/tex] that partitions the directed line segment from point [tex]\( A = (x_1, y_1) \)[/tex] to point [tex]\( B = (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], we use the section formula. The coordinates of the point [tex]\( E \)[/tex], which divides the line segment joining [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in the ratio [tex]\( m:n \)[/tex], are given by:
[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]
Given:
- [tex]\( A = (0, 1) \)[/tex]
- [tex]\( B = (1, 0) \)[/tex]
- Ratio [tex]\( m:n = 1:2 \)[/tex]
Let's plug these values into the formulas:
For the [tex]\( x \)[/tex]-coordinate:
[tex]\[
x = \left(\frac{1}{1+2}\right)(1 - 0) + 0
\][/tex]
[tex]\[
x = \left(\frac{1}{3}\right)(1) + 0
\][/tex]
[tex]\[
x = \frac{1}{3}
\][/tex]
For the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \left(\frac{1}{1+2}\right)(0 - 1) + 1
\][/tex]
[tex]\[
y = \left(\frac{1}{3}\right)(-1) + 1
\][/tex]
[tex]\[
y = -\frac{1}{3} + 1
\][/tex]
[tex]\[
y = \frac{2}{3}
\][/tex]
So, the coordinates of point [tex]\( E \)[/tex] that divides the line segment [tex]\( AB \)[/tex] in the ratio 1:2 are:
[tex]\[
\left( \frac{1}{3}, \frac{2}{3} \right) \approx (0.3333333333333333, 0.6666666666666667)
\][/tex]
Therefore, the coordinates of point [tex]\( E \)[/tex] are [tex]\( \left( 0.3333333333333333, 0.6666666666666667 \right) \)[/tex].
