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Lets say you have $\vec{a}$(2,1) and $\vec{b}$(0,3)

Are these lines the same?

$ L_1 = \{\vec{b} + t_1 (\vec{b}-\vec{a}) \mid t_1 \in\mathbb{R} \}$

$ L_2 = \{\vec{a} + t_2 (\vec{b}-\vec{a}) \mid t_2 \in\mathbb{R} \}$

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What is the definition of a line? That is helpful here. What can't you do? –  Daryl Oct 11 '12 at 20:19
    
How can I explain this.. I want to know if the span of these 2 L's in $\mathbb{R}^2$ is the same. –  ZafarS Oct 11 '12 at 20:22
    
@ZafarS "span of a line"? –  rschwieb Oct 11 '12 at 20:27
    
@ZafarS hint Can one be rearranged to be exactly like the other? –  Daryl Oct 11 '12 at 20:29
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$2$ points are sufficient to determine a line. You've got two points. If both points satisfy both lines, it means the lines are same as there is a unique line passing through two distinct points. –  TheJoker Oct 11 '12 at 20:31

1 Answer 1

up vote 1 down vote accepted

$$\begin{array}{lcl} L_2 & = &\{ a + t_2(b-a) & \mid t_2 \in\mathbb{R} \} \\ & = &\{ (b-b)+ a + t_2(b-a)&\mid t_2 \in\mathbb{R} \} \\ & = &\{b + (t_2-1)(b-a)&\mid t_2 \in\mathbb{R}\} \\ & = &\{b + t'(b-a)& \mid t' \in \mathbb{R}\} \\ & = &L_1 \end{array}$$

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