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]\( 5:1 \)[/tex], we will use the given formula for dividing a line segment in a given ratio. The formula provided is:
[tex]\[ v = \left(\frac{m}{m+n}\right)(v_2 - v_1) + v_1 \][/tex]
Here, [tex]\( m \)[/tex] and [tex]\( n \)[/tex] represent the ratios, and [tex]\( v_1 \)[/tex] and [tex]\( v_2 \)[/tex] are the coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex] respectively.
Given:
- [tex]\( m = 5 \)[/tex]
- [tex]\( n = 1 \)[/tex]
- Coordinates of [tex]\( J \)[/tex] ([tex]\( v_1 \)[/tex]) = [tex]\(-8\)[/tex]
- Coordinates of [tex]\( K \)[/tex] ([tex]\( v_2 \)[/tex]) = [tex]\( 6 \)[/tex]
We now substitute these values into the formula:
1. Calculate the fraction:
[tex]\[ \frac{m}{m+n} = \frac{5}{5+1} = \frac{5}{6} \][/tex]
2. Subtract the coordinates [tex]\( v_1 \)[/tex] from [tex]\( v_2 \)[/tex]:
[tex]\[ v_2 - v_1 = 6 - (-8) = 6 + 8 = 14 \][/tex]
3. Multiply the fraction by the difference:
[tex]\[ \left(\frac{5}{6}\right) \times 14 = \frac{5 \times 14}{6} = \frac{70}{6} \approx 11.666666666666668 \][/tex]
4. Finally, add this result to [tex]\( v_1 \)[/tex] to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[ v = 11.666666666666668 - 8 = 3.666666666666668 \][/tex]
Therefore, the [tex]\( y \)[/tex]-coordinate of the point that divides the line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio of [tex]\( 5:1 \)[/tex] is:
[tex]\[ \boxed{3.666666666666668} \][/tex]
