Question

Can someone help me to answer this please?
Let P1,P2,p3,P4 be points fixed relative to an origin O and let vectors r1,r2,r3,r4 be position vectors from O to each points. show that if the vector equation a1r1+a2r2+a3r3+a4r4=0 (here a1,a2,a3 and a4 are scalars and r1,r2,r3 and r4 are vectors) holds with respect to the origin O, then it will hold with respect to any other origin O' if and only if a1+a2+a3+a4=0.

Answer 1

If V ={x,y} is the position vector with respect O=(0,0), then
V={x-c, y-d} will be the position vector with respect O'=(c,d)
The equations with respect to O are
\[
a(1) x(1)+a(2) x(2)+a(3)x(3)+a(4) x(4)=0\\
a(1) y(1)+a(2) y(2)+a(3)
y(3)+a(4) y(4)=0
\]
The equations with respect to O' are

\[
-c(a(1)+a(2)+a(3)+ a(4)) +\\a(1)
x(1)+a(2) x(2)+a(3) x(3)+a(4)
x(4)=0 \\
\]
\[
-d(a(1) +a(2)+a(3) +a(4))+\\
a(1) y(1)+a(2) y(2)+a(3)
y(3)+a(4) y(4)=0

\]
Now you can solve your problem.