Vector proof that a quadrilateral is a rectangle under certain conditions

• May 9th 2013, 05:12 AM
deSitter
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,
• May 9th 2013, 06:24 AM
emakarov
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.
• May 9th 2013, 07:10 AM
zhandele
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.
• May 9th 2013, 06:11 PM
Soroban
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.

Quote:

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

Quote:

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

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

Quote:

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

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

We have only this:
Code:

      *       *     c *       *       *  *  *  *           a

Quote:

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

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

The diagram could look like this:
Code:

      *       *          *       c *          * b         *        *           *  *  *  *               a

Quote:

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

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

The diagram could look like this:
Code:

      *       *     c *          *       *          * b       *          *       *  *  *  *  *             a
• May 9th 2013, 10:21 PM
deSitter
Re: Vector proof that a quadrilateral is a rectangle under certain conditions
Thank you very much, you have all been very helpful.