To find the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 2:3 \)[/tex], we will use the section formula. Here's a detailed, step-by-step solution to solve the problem:
Given:
- The coordinates of point [tex]\( J \)[/tex] (let's call it [tex]\( v_1 \)[/tex]) are [tex]\( v_1 = -6 \)[/tex].
- The coordinates of point [tex]\( K \)[/tex] (let's call it [tex]\( v_2 \)[/tex]) are [tex]\( v_2 = 7 \)[/tex].
- The ratio [tex]\( m:n = 2:3 \)[/tex].
We need to find the [tex]\( y \)[/tex]-coordinate of the point [tex]\( P \)[/tex] that divides the line segment [tex]\( JK \)[/tex] in the ratio [tex]\( 2:3 \)[/tex].
### Step-by-Step Solution:
1. Identify the formula for the [tex]\( y \)[/tex]-coordinate of the point that divides the line segment:
[tex]\[
v = \left(\frac{m}{m+n}\right) \left(v_2 - v_1 \right) + v_1
\][/tex]
2. Substitute the values into the formula:
- [tex]\( m = 2 \)[/tex]
- [tex]\( n = 3 \)[/tex]
- [tex]\( v_1 = -6 \)[/tex]
- [tex]\( v_2 = 7 \)[/tex]
3. Calculate [tex]\( m + n \)[/tex]:
[tex]\[
m + n = 2 + 3 = 5
\][/tex]
4. Calculate the difference [tex]\( v_2 - v_1 \)[/tex]:
[tex]\[
v_2 - v_1 = 7 - (-6) = 7 + 6 = 13
\][/tex]
5. Substitute these values into the formula:
[tex]\[
v = \left(\frac{2}{5}\right) \left(13\right) + \left(-6\right)
\][/tex]
6. Multiply the ratio with the difference:
[tex]\[
\frac{2}{5} \times 13 = \frac{26}{5} = 5.2
\][/tex]
7. Add this value to [tex]\( v_1 \)[/tex]:
[tex]\[
v = 5.2 + (-6) = 5.2 - 6 = -0.8
\][/tex]
8. Conclusion: The [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 2:3 \)[/tex] is:
[tex]\[
\boxed{-0.8}
\][/tex]
This is the [tex]\( y \)[/tex]-coordinate we were looking for.
