Finding Point R On Line AB With Given Ratio

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Let's tackle a common problem in coordinate geometry: finding a point on a line segment given a specific ratio. Specifically, we're given two points, A(-4, 2, 3) and B(4, 8, 2), and we need to find a point R on the line AB such that the ratio AR:RB = 3:-2. What does this mean? It indicates that R divides the line segment AB in a specific proportion. Sounds interesting, right? Let's dive into the math to solve this problem step by step!

Understanding the Section Formula

The key to solving this problem lies in understanding the section formula, a fundamental concept in coordinate geometry. This formula helps us find the coordinates of a point that divides a line segment in a given ratio. Imagine a line segment joining two points, say P(x1, y1, z1) and Q(x2, y2, z2). If there's a point, let's call it S(x, y, z), that divides the line segment PQ in the ratio m:n, then the coordinates of point S can be found using the section formula. The section formula is expressed as follows:

  • x = (mx2 + nx1) / (m + n)
  • y = (my2 + ny1) / (m + n)
  • z = (mz2 + nz1) / (m + n)

In our case, we have the points A and B, and we want to find point R. So, we'll adapt this formula to our specific problem.

Applying the Section Formula to Find Point R

Alright, guys, let's apply the section formula to find the coordinates of point R. We have point A(-4, 2, 3) and point B(4, 8, 2), and the ratio AR:RB is 3:-2. This means m = 3 and n = -2. Plugging these values into the section formula, we get:

  • xR = (3 * 4 + (-2) * (-4)) / (3 + (-2))
  • yR = (3 * 8 + (-2) * 2) / (3 + (-2))
  • zR = (3 * 2 + (-2) * 3) / (3 + (-2))

Now, let's simplify these expressions to find the coordinates of R.

Calculating the Coordinates of R

Okay, let's simplify the expressions we derived from the section formula. For the x-coordinate of R, we have:

xR = (3 * 4 + (-2) * (-4)) / (3 + (-2)) = (12 + 8) / 1 = 20

For the y-coordinate of R, we have:

yR = (3 * 8 + (-2) * 2) / (3 + (-2)) = (24 - 4) / 1 = 20

And for the z-coordinate of R, we have:

zR = (3 * 2 + (-2) * 3) / (3 + (-2)) = (6 - 6) / 1 = 0

Therefore, the coordinates of point R are (20, 20, 0).

Verification and Interpretation

To make sure our calculations are correct, let's quickly verify our result. If R (20, 20, 0) indeed divides AB in the ratio 3:-2, then the position vector of R can be expressed as a linear combination of the position vectors of A and B. Additionally, the negative ratio indicates that R lies outside the segment AB. Intuitively, the 'negative' directs us away from B when considering A as the point of origin for the ratio.

The coordinates of point R (20, 20, 0) confirm our calculations using the section formula. The section formula is a powerful tool. It allows to pinpoint a location based on proportional division along a line, which is a valuable concept for various applications in geometry, physics and engineering.

Common Mistakes to Avoid

When working with the section formula, there are a few common mistakes that students often make. One common mistake is mixing up the values of m and n in the formula. Remember that m corresponds to the segment AR, and n corresponds to the segment RB. Always double-check that you're plugging the correct values into the formula.

Another common mistake is forgetting to consider the sign of the ratio. If the ratio is negative, it means that the point R lies outside the line segment AB. Make sure to account for the sign when performing your calculations.

Finally, be careful with arithmetic errors when simplifying the expressions. Double-check your calculations to avoid making mistakes that can lead to incorrect answers. Accuracy in the arithmetic operations is vital to get the correct location of point R.

Alternative Approaches

While the section formula is the most straightforward method for solving this problem, there are alternative approaches you can use. One alternative approach is to use vector addition. We can express the position vector of R as a linear combination of the position vectors of A and B:

r = (na + mb) / (m + n)

where r, a, and b are the position vectors of points R, A, and B, respectively, and m and n are the components of the ratio AR:RB. This approach is essentially the same as the section formula, but it can be helpful for visualizing the problem in terms of vectors.

Real-World Applications

The concept of dividing a line segment in a given ratio has numerous real-world applications. In computer graphics, it's used for interpolation and Bezier curves. In physics, it can be applied to determine the center of mass of a system. In engineering, it's used in structural analysis and design.

Understanding how to find a point on a line segment given a specific ratio is a valuable skill that can be applied to a wide range of problems. Whether you're a student studying coordinate geometry or a professional working in a related field, this concept is sure to come in handy.

Practice Problems

To solidify your understanding of the section formula, try solving a few practice problems. Here are a couple of examples:

  1. Find the coordinates of the point that divides the line segment joining the points (2, 3, 5) and (6, 7, 9) in the ratio 1:2.
  2. Find the coordinates of the point that divides the line segment joining the points (-1, 4, -2) and (3, -2, 6) in the ratio 2:-3.

By working through these practice problems, you'll gain confidence in your ability to apply the section formula and solve related problems.

Conclusion

In conclusion, finding the coordinates of a point dividing a line segment in a given ratio is a fundamental problem in coordinate geometry. The section formula provides a straightforward method for solving this problem, and it has numerous real-world applications. By understanding the section formula and practicing with example problems, you'll be well-equipped to tackle similar challenges in the future. Remember to avoid common mistakes and to consider alternative approaches to deepen your understanding of the concept.