Sure, let's find the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( v_1 \)[/tex] to [tex]\( v_2 \)[/tex] in the ratio [tex]\( 5:1 \)[/tex].
Given:
- [tex]\( v_1 = -8 \)[/tex]
- [tex]\( v_2 = 6 \)[/tex]
- The ratio [tex]\( m:n = 5:1 \)[/tex]
We can use the section formula for finding the coordinates of the point that divides a line segment in a given ratio [tex]\( m:n \)[/tex]. The formula for the [tex]\( y \)[/tex]-coordinate is:
[tex]\[ y = \left( \frac{m}{m+n} \right)(v_2 - v_1) + v_1 \][/tex]
Let's plug in the values:
1. The ratio [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are [tex]\( 5 \)[/tex] and [tex]\( 1 \)[/tex], respectively.
2. So, [tex]\( m + n = 5 + 1 = 6 \)[/tex].
3. Then, calculate the fraction on the right-hand side: [tex]\( \frac{m}{m+n} = \frac{5}{6} \)[/tex].
4. Compute the difference [tex]\( v_2 - v_1 \)[/tex]: [tex]\( 6 - (-8) = 6 + 8 = 14 \)[/tex].
5. Multiply the fraction by the difference: [tex]\( \frac{5}{6} \times 14 = \frac{70}{6} \)[/tex].
6. Simplify [tex]\( \frac{70}{6} \)[/tex] to get approximately [tex]\( 11.6667 \)[/tex].
7. Finally, add [tex]\( v_1 \)[/tex] to this result: [tex]\( 11.6667 + (-8) \)[/tex].
Thus, performing the calculation:
[tex]\[ 11.6667 - 8 = 3.6667 \][/tex]
So, the [tex]\( y \)[/tex]-coordinate of the point that divides the line segment from [tex]\( v_1 \)[/tex] to [tex]\( v_2 \)[/tex] in the ratio [tex]\( 5:1 \)[/tex] is approximately 3.6667.
