To find the coordinates of point [tex]\( E \)[/tex] that partitions the directed line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex], we will use the section formula for internal division. Given the points [tex]\( A(0,1) \)[/tex] and [tex]\( B(-1,3) \)[/tex], and the ratio [tex]\( m:n = 1:2 \)[/tex], the coordinates of [tex]\( E \)[/tex] can be calculated as follows:
1. Identify the coordinates of points A and B:
- [tex]\( A = (x_1, y_1) = (0, 1) \)[/tex]
- [tex]\( B = (x_2, y_2) = (-1, 3) \)[/tex]
2. Use the section formula for internal division:
The section formula for a point dividing the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] is:
[tex]\[
x = \left(\frac{m}{m+n}\right) \cdot (x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right) \cdot (y_2 - y_1) + y_1
\][/tex]
3. Substitute the given values into the formula:
- For the [tex]\( x \)[/tex]-coordinate:
[tex]\[
x = \left(\frac{1}{1+2}\right) \cdot (x_2 - x_1) + x_1
\][/tex]
[tex]\[
x = \left(\frac{1}{3}\right) \cdot (-1 - 0) + 0
\][/tex]
[tex]\[
x = \left(\frac{1}{3}\right) \cdot (-1) + 0
\][/tex]
[tex]\[
x = -\frac{1}{3}
\][/tex]
- For the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \left(\frac{1}{1+2}\right) \cdot (y_2 - y_1) + y_1
\][/tex]
[tex]\[
y = \left(\frac{1}{3}\right) \cdot (3 - 1) + 1
\][/tex]
[tex]\[
y = \left(\frac{1}{3}\right) \cdot 2 + 1
\][/tex]
[tex]\[
y = \frac{2}{3} + 1
\][/tex]
[tex]\[
y = \frac{2}{3} + \frac{3}{3}
\][/tex]
[tex]\[
y = \frac{5}{3}
\][/tex]
Thus, the coordinates of point [tex]\( E \)[/tex] which divides the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 1:2 \)[/tex] are:
[tex]\[
E \left( -\frac{1}{3}, \frac{5}{3} \right)
\][/tex]
Converting these into decimal form, we get:
[tex]\[
\boxed{E \left( -0.3333333333333333, 1.6666666666666665 \right)}
\][/tex]
