A line segment starts at P(4,-7) and ends at Q(2,3). Find the parametric equations of the line segment.
the slope is -5
x = 4 - t
y = -7 + 5t
To find the parametric equations of a line segment, we need to determine the equations that describe the x and y coordinates as a function of a parameter, usually denoted as t.
Step 1: Find the direction vector of the line segment.
The direction vector is obtained by subtracting the coordinates of the starting point from the coordinates of the ending point.
Let PQ be the direction vector of the line segment.
PQ = (x2 - x1, y2 - y1)
= (2 - 4, 3 - (-7))
= (-2, 10)
Step 2: Determine the parametric equations.
The parametric equations are given by:
x = x1 + at
y = y1 + bt
where (x1, y1) are the coordinates of the starting point and (a, b) are the components of the direction vector.
Substituting the values into the equations, we have:
x = 4 + (-2)t
y = -7 + 10t
Thus, the parametric equations of the line segment are:
x = 4 - 2t
y = -7 + 10t