To find the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio of [tex]\( 5:1 \)[/tex], we can use the given formula:
[tex]\[ v = \left(\frac{m}{m+n}\right)(v_2 - v_1) + v_1 \][/tex]
Let’s break this down step-by-step:
1. Identify the values:
- [tex]\( m = 5 \)[/tex] (the first part of the ratio)
- [tex]\( n = 1 \)[/tex] (the second part of the ratio)
- [tex]\( v_1 = -8 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( J \)[/tex])
- [tex]\( v_2 = 6 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( K \)[/tex])
2. Substitute the values into the formula:
[tex]\[
v = \left(\frac{5}{5+1}\right)(6 - (-8)) + (-8)
\][/tex]
3. Simplify the ratio inside the formula:
[tex]\[
\frac{5}{5+1} = \frac{5}{6}
\][/tex]
4. Calculate the difference between [tex]\( v_2 \)[/tex] and [tex]\( v_1 \)[/tex]:
[tex]\[
v_2 - v_1 = 6 - (-8) = 6 + 8 = 14
\][/tex]
5. Multiply the ratio [tex]\(\frac{5}{6}\)[/tex] by the difference [tex]\( 14 \)[/tex]:
[tex]\[
\frac{5}{6} \times 14 = \frac{70}{6} \approx 11.6667
\][/tex]
6. Add [tex]\( v_1 \)[/tex] to the product:
[tex]\[
v = 11.6667 + (-8) = 11.6667 - 8 \approx 3.6667
\][/tex]
Hence, the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio of [tex]\( 5:1 \)[/tex] is approximately [tex]\( 3.6667 \)[/tex].
