IDNLearn.com: Your trusted source for accurate and reliable answers. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.
Sagot :
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.
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.