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$(A,B)$ and $(C,D)$ are parallel vectors, in the book I'm reading, it illustrates one case for this proposition: $(A,B)\sim (C,D) \implies (A,C)\sim (B,D)$ with the following figure:

enter image description here

And then there's an exercise asking me to draw that proposition in the case that $A,B,C,D$ are collinear. I'm not sure about the answer but I guess that to achieve that, I should think of null vectors. The only possibility I see to achieve collinearity between all them is to think of $A=B=C=D$. Is this the answer?

Edit: I guess this is a viable answer:

enter image description here

The vector $(A,B)$ approach to $(C,D)$ until one is over the other.

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1  
What are you talking about? –  copper.hat Dec 13 '12 at 17:43
    
Is it better now? –  Vÿska Dec 13 '12 at 18:44
    
Excuse the holes in my education, but what does ~ mean? –  copper.hat Dec 13 '12 at 18:46
1  
@copper.hat $\sim$ means similar. About the holes on your education, there are only two reasons to be here: 1 - Filling these holes with knowing; 2 - Help others to do so. –  Vÿska Dec 13 '12 at 18:56
    
What other rules are satisfied? Is $(A,B)+(B,D) \sim (A,D)$? –  copper.hat Dec 13 '12 at 18:59

1 Answer 1

Ok, so correct me if I'm wrong, but I assume by collinear that just means points A, B, C, and D all fall on the same line.

So, assume they all fall on a straight line:

enter image description here

Suppose vector AB is magnitude +3, similar to CD. Then suppose on the line A,B,C, and D fall on there's X magnitude between BC. It follows that AC is 3+X and BD is 3+X, hence AC~BD?


From Wikipedia:

In geometry, collinearity is a property of a set of points, specifically, the property of lying on a single line.

In other words, a bunch of points that fall on the same line, e.g. are aligned:

enter image description here

Now if A, B, C, and D are collinear (as above), and the distance (magnitude) of $\overrightarrow {AB}$ is the same (~) as $\overrightarrow {CD}$ then you can see, visually that it follows that $\overrightarrow {AC}$ ~ $\overrightarrow {BD}$

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Trying to understand. Did you see the edit in my post? –  Vÿska Dec 15 '12 at 14:02
    
But shouldn't you draw a case in which $A,B,C,D$ are collinear? Is this the case? –  Vÿska Dec 15 '12 at 16:11
    
I added more info. –  user17753 Jan 11 '13 at 16:20

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