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 $\alpha\beta>0$ then $\alpha\vec{a}+\beta\vec{b}$ is a multiple of $\vec{a}+\vec{b}$.

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

So?

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