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Chapter 11: Problem 46

Find the point on the line segment joining \(P_{1}(1,4,-3)\) and \(P_{2}(1,5,-1)\) that is \(\frac{2}{3}\) of the way from \(P_{1}\) to \(P_{2}\)

Short Answer

Expert verified
The point is \( \left(1, \frac{14}{3}, -\frac{5}{3}\right) \).

Step by step solution

01

Express the Segment with a Parametric Equation

A point on the line segment can be described using a parametric equation: \( P(t) = (1-t) \, P_{1} + t \, P_{2} \), where \( 0 \leq t \leq 1 \), \( P_1 = (1,4,-3) \), and \( P_2 = (1,5,-1) \).
02

Determine the Parameter Value

Since we need a point that is \( \frac{2}{3} \) of the way from \( P_{1} \) to \( P_{2} \), we set \( t = \frac{2}{3} \).
03

Plug the Parameter into the Parametric Equation

Substitute \( t = \frac{2}{3} \) into the equation: \[ P(t) = \left(1-\frac{2}{3}\right) (1,4,-3) + \frac{2}{3}(1,5,-1) \]
04

Compute the Coordinates

Calculate each coordinate using the expression derived in Step 3:\[ x = \left(1-\frac{2}{3}\right) \times 1 + \frac{2}{3} \times 1 = 1 \]\[ y = \left(1-\frac{2}{3}\right) \times 4 + \frac{2}{3} \times 5 = \frac{4}{3} + \frac{10}{3} = \frac{14}{3} \]\[ z = \left(1-\frac{2}{3}\right) \times (-3) + \frac{2}{3} \times (-1) = -1 + (-\frac{2}{3}) = -\frac{5}{3} \]
05

Identify the Point on the Segment

The coordinates at \( t = \frac{2}{3} \) give the point \( P = (1, \frac{14}{3}, -\frac{5}{3}) \). So, the point that is \( \frac{2}{3} \) of the way from \( P_{1} \) to \( P_{2} \) is \( P = \left(1, \frac{14}{3}, -\frac{5}{3}\right) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric equations
Parametric equations are a powerful tool used in calculus to express the coordinates of the points on a curve or line as functions of a single variable, typically denoted as \( t \). This approach is particularly useful when dealing with line segments, as it allows us to calculate any intermediate point along the segment by simply substituting a value for \( t \).
  • The general form of a parametric equation for a line segment between two points \( P_1 = (x_1, y_1, z_1) \) and \( P_2 = (x_2, y_2, z_2) \) is:
\[ P(t) = (1-t) \, P_1 + t \, P_2 \]
  • Where \( 0 \leq t \leq 1 \), \( t \) represents how far along the segment the point is.
  • When \( t = 0 \), the equation yields point \( P_1 \), and when \( t = 1 \), it yields point \( P_2 \).
  • This allows for smooth transitions between these two endpoints.

Overall, parametric equations simplify the process of finding specific points along a line segment by encoding the entire segment in a single formula.
Line segment
A line segment represents the shortest path connecting two points in space, and it has distinct properties that differ from an entire line.
  • Unlike a full line, which extends infinitely in both directions, a line segment has distinct start and end points.
  • For the segment discussed in our example, we connect points \( P_1 = (1,4,-3) \) and \( P_2 = (1,5,-1) \).
  • This particular segment lies in a three-dimensional space as indicated by its coordinates.

Visualizing a line segment can help in understanding the spatial relationship between points. In our context, using a parametric equation, we can pinpoint any location along the segment based on a parameter \( t \). This is especially useful in problems requiring finding a specific division, as it allows us to handle the calculations analytically and flexibly.
Coordinate calculation
Calculating the coordinates for a specific point on a line segment involves some arithmetic but is straightforward when using parametric equations. Here鈥檚 the breakdown of how you can determine the desired coordinates:
  • First, identify the parameter \( t \) for the point's position relative to the two endpoints.
  • Substitute the value of \( t \) into the parametric equation.
  • For our example, \( t = \frac{2}{3} \) means we are looking \( \frac{2}{3} \) of the way from \( P_1 \) to \( P_2 \).
  • Compute each coordinate separately:
    • The x-coordinate remains constant due to both points having the same x-value: \( x = 1 \).
    • For the y-coordinate, calculate: \( y = (1-\frac{2}{3}) \times 4 + \frac{2}{3} \times 5 = \frac{14}{3} \).
    • For the z-coordinate, calculate: \( z = (1-\frac{2}{3}) \times (-3) + \frac{2}{3} \times (-1) = -\frac{5}{3} \).

With those calculations, the coordinates for the point \( \frac{2}{3} \) of the way from \( P_1 \) to \( P_2 \) are \( (1, \frac{14}{3}, -\frac{5}{3}) \). By following this method, you can easily find exact positions along any line segment using parametric equations.

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