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Math Help - Vector proof that a quadrilateral is a rectangle under certain conditions

  1. #1
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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.

    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?


    Vector proof that a quadrilateral is a rectangle under certain conditions-quadrilateral.jpg

    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,
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  2. #2
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    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.
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  3. #3
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    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.
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  4. #4
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    Re: Vector proof that a quadrilateral is a rectangle under certain conditions

    Hello, deSitter!

    Since dot products are used,
    . . I assume that all line segments are vectors.


    I am required not only to prove the correct answer,
    but also to state and show mathematically why the others are not correct.

    In quadrilateral OABC\!:\;\overrightarrow{OA} = \vec a,\,\overrightarrow{AB} = \vec b,\,\overrightarrow{OC} = \vec c.

    For which one of the following sets of conditions must OABC be a rectangle?

    Code:
        C o
           *    *
            *         *
             *              *      B
              *                   o
             c *                 *
                *               * b
                 *             *
                  o  *  *  *  o
                 O      a      A
    (A)\;\vec a\cdot \vec c = 0\text{ and }\vec b = \vec c

    This is the one.

    We have: . \vec a \perp \vec c\,\text{ and }\,\vec b \parallel \vec c\,\text{ and }\,|\vec b| = |\vec c|

    The diagram looks like this:
    Code:
          *           *
        c *           * b
          *           *
          *  *  *  *  *
                a



    (B)\;\vec a\cdot \vec a = \vec c \cdot \vec c \,\text{ and }\,\vec b = \vec c

    We have: . |\vec a| = |\vec c|\,\text{ and }\,\vec b \parallel \vec c\,\text{ and }\,|\vec b| = |\vec c|

    The diagram could look like this:
    Code:
          *        *
           *        * b
          c *        *
             *        *
              *  *  *  *
                   a



    (C)\;\vec a \cdot \vec c \,=\,0\,\text{ and }\,\vec a\cdot \vec a\,=\,\vec c\cdot\vec c

    We have: . \vec a \perp \vec c\,\text{ and }\,|\vec a| = |\vec c|

    We have only this:
    Code:
          *
          *
        c *
          *
          *  *  *  *
               a



    (D)\;\vec a \cdot \vec a = \vec c \cdot \vec c \,\text{ and }\,\vec a\cdot \vec b = 0

    We have: . |\vec a| = |\vec c|\,\text{ and }\,\vec a \perp \vec b

    The diagram could look like this:
    Code:
          *
           *           *
          c *          * b
             *         *
              *  *  *  *
                   a



    (E)\;\vec a \cdot\vec b\,\text{ and }\,\vec a \cdot \vec c = 0

    We have: . \vec a \perp \vec b\,\text{ and }\,\vec a \perp \vec c

    The diagram could look like this:
    Code:
          *
          *
        c *           *
          *           * b
          *           *
          *  *  *  *  *
                a
    Thanks from deSitter
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  5. #5
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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.
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