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Vector proof that a quadrilateral is a rectangle under certain conditions

Hello,

I'm having some issues with the following multiple choice question. I am required not only to prove the correct answer, but also to state and show mathematically why the others are not correct.

Quote:

*In the quadrilateral OABC, OA = a, AB = b and OC = c (a, b, c ≠ 0). For which one of the following sets of condition must OABC be a rectangle?*

Attachment 28308

A. a.c = 0 and b = c

B. a.a = c.c and b = c

C. a.c = 0 and a.a = c.c

D. a.a = c.c and a.b = 0

E. a.b = 0 and a.c = 0

Response A appears to allow for a rectangle, but also a square. While response B appears to be just a square. The remaining 3 responses do not seem to be limited to being only a rectangle. How would you suggest I tackle this problem. My observations are not really backed up by any substantial math. Which response allows for **only** a rectangle and how would you go about proving this.

Cheers,

Re: Vector proof that a quadrilateral is a rectangle under certain conditions

In response A, the fact that b = c (as vectors) implies that the quadrilateral is a parallelogram and the fact that the dot product of a and c is zero implies that one of the angles is right, i.e., the quadrilateral is a rectangle (not necessarily a square). In response B, the quadrilateral is a parallelogram with a = c (equal as lengths, not as vectors). This is a rhombus, but not necessarily a rectangle because nothing is said about the angles.

Re: Vector proof that a quadrilateral is a rectangle under certain conditions

Do you have a geometry program, like Geometer's Sketchpad or Cabri or Geogebra? The latter is free, and in some ways the best of the three.

If a.a = c.c, then a and c are the same length. If they both start at O, their end points are on a circle centered at O, but you don't know anything about the angle between them. In one of these programs, you could draw a circle, construct two points on it, and drag them around. It might give you more insight.

Quick comment on e). We have two right angles there (make sure you understand why a zero dot product means a right angle; some books define perpendicularity in terms of the dot product). But we don't know anything about the lengths of b and c.

Re: Vector proof that a quadrilateral is a rectangle under certain conditions

Hello, deSitter!

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Re: Vector proof that a quadrilateral is a rectangle under certain conditions

Thank you very much, you have all been very helpful.