(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 $0< t <1,$ 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 $0< t <1,$ 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 $(3,-1).$

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Key Concepts

Endpoint Evaluation

Evaluating the parametric equations at the extreme values of the parameter—typically t = 0 and t = 1—retrieves the endpoints of the segment. This verification ensures that the parametric form correctly represents the finite segment connecting the two points, rather than extending infinitely.

Direction Vector

The direction vector is obtained by subtracting the coordinates of the initial point from the terminal point. It defines the orientation and magnitude of the segment, and when multiplied by the parameter t and added to the starting point, it constructs the line segment in space.

Linear Interpolation

Linear interpolation is the process of finding a point along the straight line between two fixed points by mixing their coordinates based on a parameter. When the parameter t varies between 0 and 1, every point on the segment can be represented as a weighted average of the endpoints, where t determines the proportional distance from one endpoint to the other.

Parametric Equations for Lines

Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted as t. This form captures the entire line through a continuous mapping from the parameter t to points in the plane, allowing for a clear description of the change in position with respect to t.

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