(a) Show that the parametric equations x=x1+(x2-x1) t y=y1+(y2-y1) t where 0 ⩽t ⩽1, describe t

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(a) Show that the parametric equations x=x1+(x2-x1) t ...

(a) Show that the parametric equations $$x=x_{1}+\left(x_{2}-x_{1}\right) t \quad y=y_{1}+\left(y_{2}-y_{1}\right) t$$ where $0 \leqslant t \leqslant 1,$ describe the line segment that joins the points $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right) .$ (b) Find parametric equations to represent the line segment from $(-2,7)$ to $(3,-1)$ .

(a) Show that the parametric equations
$$x=x_{1}+\left(x_{2}-x_{1}\right) t \quad y=y_{1}+\left(y_{2}-y_{1}\right) t$$
where $0 \leqslant t \leqslant 1,$ describe the line segment that joins
the points $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right) .$
(b) Find parametric equations to represent the line segment
from $(-2,7)$ to $(3,-1)$ .

Key Concepts

Parametric Equations

Parametric equations express the coordinates of the points on a geometric object, such as a line or curve, as functions of one or more independent parameters. This method allows one to describe the entire object as the parameter varies over a specified interval, providing a flexible and dynamic representation in contrast to a single algebraic equation.

Line Segment

A line segment is a part of a line that is bounded by two endpoints. In analytic geometry, the line segment can be represented by constraining the parameter used in its parametric equations to an interval, ensuring that only the points between the endpoints are considered. This constrains the solution to a finite set of points lying along the straight path connecting the endpoints.

Linear Interpolation

Linear interpolation is the method of finding a point between two known points along a straight line. It is achieved by taking a weighted average of the two endpoints with weights that depend on a parameter, typically denoted as t. When t varies from 0 to 1, the interpolated points transition linearly from the first endpoint to the second, effectively mapping out the entire line segment.

Convex Combination

A convex combination involves expressing a point as the weighted sum of two endpoints where the weights are non-negative and add up to one. This concept is central to parametric representations of line segments because it guarantees that the resulting point lies within the segment between the two endpoints. When t ranges from 0 to 1, the point is a convex combination of the endpoints, ensuring it remains within the boundaries defined by them.

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