Find parametric equations for the line segment joining...

Find parametric equations for the line segment joining
$P(2,-1)$ and $Q(3,1),$ and use the result in Exercise 25 to find
(a) the midpoint between $P$ and $Q$
(b) the point that is one-fourth of the way from $P$ to $Q$
(c) the point that is three-fourths of the way from $P$ to $Q$.

Key Concepts

Linear Interpolation and Convex Combinations

Linear interpolation involves finding a point between two given points by taking a weighted average of the coordinates. This method is based on the concept of convex combinations, where the weights are non-negative and add up to one. By varying the weights, specific points like the midpoint or other fractionally-distanced points can be computed, allowing for a straightforward way to partition the line segment.

Application of Parametric Representation in Geometry

Using parametric equations allows for a systematic approach to problems in coordinate geometry, such as dividing a line segment into specific ratios. By substituting appropriate parameter values (for instance, t = 0.25, 0.5, and 0.75), one can easily determine points that lie at given fractional distances between the endpoints, thereby simplifying the process of calculating midpoints or other internal division points.

Parametric Equations of a Line

Parametric equations represent a line by expressing its coordinates as functions of a parameter, usually denoted as t. For a line segment between two points, the parameter is typically restricted to a range (often 0 ? t ? 1), with t = 0 corresponding to one endpoint and t = 1 corresponding to the other. This representation makes it easy to generate any point along the segment by adjusting t.

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