Vector algebra

Question

If DEF are midpoints of triangle ABC PROVE OA+ OB+ OC =OD +OE +OF

Answer 1

Hi Saurav,

 

I'm going to assume that OX means the vector from some arbitrary origin point O to X.

 

To get halfway from A to B, you can get to A (OA) and then move halfway along the vector from A to B given by OB-OA.  That would be

OA + ½(OB-OA) = OA + ½OB- ½OA

                        = ½OA + ½OB

 

 

if DEF are midpoints of AB, BC, and CA respectively, they are given by 

 

OD = ½OA + ½OB

OE = ½OB + ½OC

OF = ½OA + ½OC

 

Adding them up, we have

 

OD + OE + OF = (½OA + ½OB) + (½OB + ½OC) + (½OA + ½OC)

               = ½OA + ½OA + ½OB + ½OB + ½OC + ½OC

               = OA + OB + OC

 

I hope that helps.

 

Regards,

Mike N.