The question asks me to find the vector form and parametric equations of the line y = 3x – 1. So what I did was the following in order:
- Found two points on the line: (1, 2) and (0, -1)
- Found the direction vector by doing (0, -1) – (1, 2) which = [-1 -3]
- I know the vector form is x = p + td, p being a point on the line and d being a direction vector so I put it in the following form:
[x y] = [ 1 2 ] + t [ -1 -3]
and so the parametric equations are as follows:
x = 1 – t
y = 2 – 3t
However, the book says the official answer is
[x y] = [ 0 -1 ] + t [ 1 3]
and then the parametric
x = t
y = -1 + 3t
I feel as if both answers are correct and I don't know how I was supposed to come to their answer instead of mine. Any help figuring out how I am supposed to reach their answer over mine either algebraically or geometrically is greatly appreciated.
Best Answer
There’s no such thing as “the” parametric or vector form of the equation of a line. As you’ve found, there can be many equivalent equations that have the correct form. In this case, the book’s answer uses a different starting point from yours, and its direction vector is in the opposite direction from the one you chose. Both your equations and the book’s describe the same line, however, so I would certainly call your solution correct.
Perhaps the book you’re using has some convention for or describes some additional constraints on what it considers “the” vector and parametric forms of a line’s equations, or perhaps it describes a specific procedure that you’re supposed to follow to get the answers in the form that the authors expect. For instance, there’s an widely-followed but tacit convention to prefer a direction vector in the first quadrant over one in the third. (Excessive minus signs are so declassé, don’t you know.) That doesn’t make the equations that you derived “wrong,” though.