Vectors in a tetrahedron?

Hi there,





The tetrahedron OABC is such that OA=a, OB=b and OC=c. The point D is the midpoint of BC and E is the midpoint of OA. The line DE is a bimedian of the tetrahedron.





Find the position vector of G, the midpoint of DE (i.e. the vector OG) and show that G is also the midpoint of the other two bimedians of the tetrahedron.


This question has got me stumped. I did Vectors a long time ago at school but nothing about bimedians.


Any help would be warmly appreciated.


Thanks

Vectors in a tetrahedron?
I'm going to give you 2 methods.





Method 1: Pure geometry





* E is the midpoint of [OA]. That means that, vectorially, OE=1/2OA.





* D is the midpoint of [BC]. Therefore, from what you know from vector operations you will have OB+OC=2OD, so OD=1/2(OB+OC) ("parallelogram rule")





* Now G is the midpoint of [DE]. So, again, just as above, you should have the relation: OD+OE=2OG, so OG=1/2(OD+OE).





But you know how to expres OD and OE in terms of OA, OB and OC: OG=1/2(1/2OA+1/2(OB+OC))





Thence, OG=1/4(OA+OB+OC). From there on, it should be straitforward to show that G is indeed the midpoint of the other 2 bimedians. Define midpoints for AB, OC and AC, and simply repeat what we did above.





Method 2: Using a coordinate system





Define a coordinate system where O is the origin, OA is the x axis, OB is the y axis, OC is the z axis. Then, algebraically, we have:





OA=%26lt;a, 0, 0%26gt;


OB=%26lt;0, b, 0%26gt;


OC=%26lt;0, 0, c%26gt;





E is the midpoint of OA, so OE=%26lt;a/2, 0, 0%26gt;





D is the midpoint of BC. So OD=1/2(OC+OC)=%26lt;0, b/2, c/2%26gt;





Now G is the midpoint of DE, so again you have OG=1/2(OD+OE)





So OG=%26lt;a/4, b/4, c/4%26gt;





So just use whichever method you prefer to finish the exercise.
Reply:Midpoint of line joining ends of two vectos (x, y) is (x+y)/2.


Remember it is vector addition.





OA = a


OB = b


OC = c





OD = (OB + OC)/2 = (b+c)/2 (midpoint of BC)


OE = (OO + OA)/2 = (0+a)/2 = a/2 (midpoint of OA)





OG = (OD + OE)/2 = [(b+c)/2 + a/2]/2 = (a+b+c)/4





For the proof that G is the midpoint of the other two bimedias:


Show that the other two bimedians give the same result.

garden centre