To find the parametric form of a line through two given points, we need to calculate the direction vector between the two points and then use that direction vector to establish the parametric form.
To calculate the direction vector, simply subtract the coordinates of the first point from the coordinates of the second point.
Given points A(5,-2) and B(8,2), the direction vector AB can be calculated as follows:
AB = B - A
So subtract the x-coordinate of A from the x-coordinate of B, and subtract the y-coordinate of A from the y-coordinate of B.
AB = [8 - 5, 2 - (-2)]
AB = [3, 4]
Next, to establish the parametric form of the line, use the direction vector in conjunction with one of the points. One commonly used form is:
r = A + t(AB)
The parameter t is a real number which allows us to generate all points along the line. When t = 0, we are at point A, and when t = 1 we are at point B. Other values of t will give us different points along the line.
For our given points and direction vector, the parametric form of the line is:
r = [5 -2] + t[3, 4]
So to recap, our direction vector is [3, 4] and the parametric form of the line passing through points (5,-2) and (8,2) is r = [5 -2] + t[3, 4]. This line passes through all points between and beyond (5,-2) and (8,2).