1. ## Vector proof

Using vectors and vector rules i need to prove that OA.OC=OB.OD given ABCD is a rectangle and O (origin) is on a different plane.

ive let OA = a, OB = b etc and drawn a diagram. Im not very good at vector proofs and dont know how to tackle the problem.

2. Originally Posted by kumquat
Using vectors and vector rules i need to prove that OA.OC=OB.OD given ABCD is a rectangle and O (origin) is on a different plane.

ive let OA = a, OB = b etc and drawn a diagram. Im not very good at vector proofs and dont know how to tackle the problem.
Dear kumquat,

$\displaystyle OA=OB+BA$

$\displaystyle OC=OB+BC$

$\displaystyle OA.OC=\left(OB+BA\right).\left(OB+BC\right)$

$\displaystyle \Rightarrow OA.OC=OB^2+OB.BC+OB.BA+BA.BC$

Since, $\displaystyle A\hat{B}C=90^{0}\Rightarrow BA.BC=0}$

$\displaystyle \Rightarrow OA.OC=OB^2+OB.BC+OB.BA$------------(1)

$\displaystyle OB.OD=OB.\left(OB+BC+CD\right)$

$\displaystyle \Rightarrow OB.OD=OB.\left(OB+BC+BA\right)$

$\displaystyle \Rightarrow OB.OD=OB^2+OB.BC}+OB.BA$-------------(2)

By (1) and (2);

$\displaystyle OA.OC=OB.OD$