Parameterization help?

Question

Find a parameterization for the line segment with initial point (5,3) terminal point (7,11) and with 0≤t≤1

Any help would be appreciated. Please explain very completely, as I never learned how to do any of this. 10 points to the best answer and thanks so much!

Answer 1

The vector from the point (5, 3) to (7, 11) is (7 - 5, 11 - 3), i.e. (2,8)

The vector from the point (5, 3) to any other point on the line will be a multiple of (2, 8). (Think of it as an arrow that points along the line and can get longer or shorter.) The 'parameter' (t) is the value of that multiple, i.e. each possible value of t corresponds to a particular point on the line.

You specified 0 < t < 1 so we have to make t = 0 correspond to the point (5, 3) and t = 1 correspond to the point (7, 11). We achieve this by making the parametric equation simply

(x, y) = (5, 3) + t(2, 8)

or x = 5 + 2t and y = 3 + 8t

If you substitute t = 0 you get (x, y) = (5, 3) + 0(2, 8) = (5, 3)

If you substitute t = 1 you get (x, y) = (5, 3) + 1(2, 8) = (5+2, 3+8) = (7, 11)

t = 0 corresponds to the initial point (5, 3)

t = 1 corresponds to the terminal point (7, 11)

Any value of t between 0 and 1 will correspond to an interior point.

Hope that explains it.