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!

Re: Proving this Parametric Equation of a Line

Quote:

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 $\displaystyle t_1\cdot t_2=0$