# Question about a scetch in the book and vector length?

The book says that RQ and SP are the same. On the drawing I can see that the length is not the same. If two vectors are equal shouldn't the length be equal too?

The book also says that AC is twice the length of both PS Sand QR. How can this be correct? I see that all three vectors are parallel.

The entire task is to express PS, AC, and QR using a b and c.

Translation: The figure shows a square OABC, where P, Q, R, and S is the midpoints on the sides OA, AB, BC and CO. We put OA= a, OB = b, OC = c. Find the vectors PS, AC and QR expressed with a, b, and c. What can we tell about the lines PS, AC and QR?

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@DylanMoreland it is part of the solution of the problem to learn a foreign language :-) I share your suspicion. –  user20266 Jan 3 '12 at 18:55
Done. And yes they are. –  Algific Jan 3 '12 at 18:56
@Thomas Well, I tried to read it in French. That didn't go well :) –  Dylan Moreland Jan 3 '12 at 18:56
@Algific Thanks! So it's a square? That certainly isn't obvious from the picture. I seem to recall geometry textbooks with purposefully misleading pictures, so that one was forced to use other arguments. –  Dylan Moreland Jan 3 '12 at 18:57
Is "firkant" necessarily a square? In Swedish, the corresponding word "fyrkant" is also used for general quadrilaterals. –  Hans Lundmark Jan 3 '12 at 21:41

Forgetting vectors for a moment and concentrating on line lengths, the question says $P,Q,R,S$ are the respective midpoints of $OA,AB,BC,CO$.

So using similar triangles comparing $OAC$ and $OPS$ and comparing $BAC$ and $BQR$ (as you say, the segments are parallel), it should be easy to see $PS$ and $QR$ are each half the length of $AC$.

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If measure the distance in the book PS and QR are not equal. Or is the figure an illustration in the sense that it is not the right ratios? –  Algific Jan 3 '12 at 19:01
It seems to me that $R$ and $Q$ may be placed incorrectly. This may be an intentional decision for the reason I gave. Draw your own figure! –  Dylan Moreland Jan 3 '12 at 19:07
If a draw my own perfect square it makes a little more sense. –  Algific Jan 3 '12 at 19:35
@Algific: Someone once defined geometry as "the science of drawing correct conclusions from incorrectly drawn figures." Don't be guided by the figure, which is likely to be inexact: be guided by the underlying mathematics. –  Arturo Magidin Jan 3 '12 at 19:56