# Thread: Proving this Parametric Equation of a Line

1. ## Proving this Parametric Equation of a Line

Hi,

I`m having trouble proving these statements involving a linear equation:

In R_n, given x = a + t*(b-a) where t is an element of [0,1] prove that:

Selecting any two values t1 and t2, the vectors connecting the two points are parallel.

The Line segment does not extend beyond points a and b.

Thanks a lot!

2. ## Re: Proving this Parametric Equation of a Line

Originally Posted by AKTilted
proving these statements involving a linear equation:
In R_n, given x = a + t*(b-a) where t is an element of [0,1] prove that:
Selecting any two values t1 and t2, the vectors connecting the two points are parallel.
The Line segment does not extend beyond points a and b.
Does this make sense to you?
If $\displaystyle \alpha\beta>0$ then $\displaystyle \alpha\vec{a}+\beta\vec{b}$ is a multiple of $\displaystyle \vec{a}+\vec{b}$.

If so then note that $\displaystyle \vec{a}+t_1(\vec{b}-\vec{a})=(1-t_1)\vec{a}+t_1\vec{b}$
Likewise $\displaystyle \vec{a}+t_2(\vec{b}-\vec{a})=(1-t_2)\vec{a}+t_2\vec{b}$

So?

P.S. As stated it may be false if $\displaystyle t_1\cdot t_2=0$