Exercise 11.  Show that, if  ,  the four points    are the vertices of a rectangle with its center at the origin.

Solution 11.

See text and/or instructor's solution manual.

Let  .  Then   ,  ,  and    .

The line segment from   to   is perpendicular to the line segment from   to ,  because

the vector from   to   is  ,  and

the vector from   to   is  ,  

and the dot product of these is clearly zero.  

A similar argument works for the other line segments.  It is also easy to show that the diagonals intersect at the origin, establishing symmetry there.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell