# Find a parametrization of the vertical line passing through the point

1. Sep 24, 2012

### Colts

1. The problem statement, all variables and given/known data
Find a parametrization of the vertical line passing through the point (7,-4,2) and use z=t as a parameter.

2. Relevant equations
r(t) = (a,b,c) + t<x,y,z>

3. The attempt at a solution
I used (7,-4,2) as (a,b,c) (the point) and used <0,0,1> for the vector since it had to be vertical and got the components to be
x=7
y=-4
z=2+t

2. Sep 24, 2012

### voko

You might as well set z = t, it amounts to the same thing.

3. Sep 24, 2012

### Colts

Ok, that's the right answer. Thank you
Why would what I did not be right?

4. Sep 24, 2012

### voko

What you did is not wrong; but the additive constant is redundant. Any mathematical expression can be written in a variety of ways with redundant terms, but such things are normally not done (unless used in some clever transformation). So you should always try to find the most concise form possible.

5. Sep 24, 2012

### HallsofIvy

Staff Emeritus
x= 7, y= 4,z= 2+ t has t=0 at (7, 4, 2) while x= 7,y= 4, z= t has t= 0 at (7, 4, 0), but they give the same line. Notice that, in the first set of equations, t= -2 gives (7, 4, 0) while, in the second set, t= 2 gives (7, 4, 2). Since a line is determined by two points, the lines defined by those sets of equations are the same line.

In fact, as long as f(x) is a one-to-one function that maps the real numbers onto the real numbers,
x= 7, y= 4, z= f(t) are parametric equations for that line.