Ok this is the last one I promise! It's from a sample exam and I'm practicing for my finals :)
Verify if the following 4 points are consecutive vertices of a parallelogram:
A(1,-1,1); B(3,0,2);C(2,3,4);D(0,2,3)
(b) Find an orthogonal vector to the plane containing the quadrilateral. What geometric interpretation can be given to the norm of the vector you just found?
Thanks!
if ABCD is a parallelogram, then
AB || DC and AD || BC
directions:
AB = (2,1,1)
BC = (-1,3,2)
DC = (2,1,1)
AD = (-1,3,2)
So, the lines are parallel as needed. Now, how about the lengths of the sides? I think that since the sides are parallel, the lengths will also have to fall into line. In fact, since the lengths are calculated using the same quantities as the slopes, they must be equal in pairs.
Pick any two adjacent side vectors, and find the cross-product.:
ABxAD = (2,1,1)x(-1,3,2) = (-1,-5,7)
The length of the vector is the area of the parallelogram.