To find the coordinates of point [tex]\(E\)[/tex] which partitions the directed line segment from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] into a ratio of [tex]\(1:2\)[/tex], we can use the section formula. The coordinates of point [tex]\(A\)[/tex] are [tex]\((0, 1)\)[/tex] and the coordinates of point [tex]\(B\)[/tex] are [tex]\((1, 0)\)[/tex]. The ratio [tex]\(m:n\)[/tex] is [tex]\(1:2\)[/tex].
The formula to find the coordinates of the point that divides the line segment in the ratio [tex]\(m:n\)[/tex] is:
[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]
Let's apply this step by step:
1. Identify the given coordinates and the ratio:
[tex]\(x_1 = 0\)[/tex], [tex]\(y_1 = 1\)[/tex]
[tex]\(x_2 = 1\)[/tex], [tex]\(y_2 = 0\)[/tex]
[tex]\(m = 1\)[/tex], [tex]\(n = 2\)[/tex]
2. Calculate the [tex]\(x\)[/tex]-coordinate:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
[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]
3. Calculate the [tex]\(y\)[/tex]-coordinate:
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
[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]\[
y = 1 - \frac{1}{3}
\][/tex]
[tex]\[
y = \frac{3}{3} - \frac{1}{3}
\][/tex]
[tex]\[
y = \frac{2}{3}
\][/tex]
Therefore, the coordinates of point [tex]\(E\)[/tex] that partitions the line segment from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] in the ratio [tex]\(1:2\)[/tex] are:
[tex]\[
(x, y) = \left( \frac{1}{3}, \frac{2}{3} \right)
\][/tex]
In decimal form, these coordinates are approximately:
[tex]\[
(x, y) \approx (0.3333333333333333, 0.6666666666666667)
\][/tex]
Thus, the [tex]\(x\)[/tex]-coordinate of point [tex]\(E\)[/tex] is [tex]\(0.3333333333333333\)[/tex] and the [tex]\(y\)[/tex]-coordinate is [tex]\(0.6666666666666667\)[/tex].
