Find a parametrization of the vertical line passing through the point

[Colts]

Homework Statement

Find a parametrization of the vertical line passing through the point (7,-4,2) and use z=t as a parameter.

Homework Equations

r(t) = (a,b,c) + t<x,y,z>

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

 

[voko]

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

 

[Colts]

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

 

[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.

 

[HallsofIvy]

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.