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okay so they want u to use a slope so, im gonna guess jut a constant slope and it is a line connecting through these points
lets start at point (-1,3) what slope do you need to hit point (3,-2)?
okay so this means that for 1 unit of x you have -5/4 units in y
we are start at -1,3 so we need to make sure x=-1 and y=3 at starting so at t=0
<-1+t,3+t> <---- is a start now we have to make sure the slope condition is satisfied for 1 change in x, we have -5/4 change in y so vector in form
this is only 1 representation of this vector, there are multiple other ways of saying this same thing, to see if they all mean the same thing, you have to satisfy 2 things, if the direction of travel is the same (i.e same unit vector direction) and that there is a common point between the other representations
because what u have defined here is not just a way to get from point -1,3 to 3,-2, but depending on whatever t you pick you have defined an infinite line
just that saying this vector this way is a convenient form as you have the starting point at t=0 and ending point at t=1
but they will ask you questions like how do you get to the ending point when t=3 or some other number and starting point at different times and such
okay that makes sense