Suppose the coordinates of n particles of masses `m_(1),m_(2),…,m_(n)` are `(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),…,(x_(n),y_(n),z_(n))` respectively.
`therefore` The position of centre of mass,
`(x,y,z)=(m_(1)(x_(1),y_(1),z_(1))+m_(2)(x_(2),y_(2),z_(2))+...+m_(n)(x_(n),y_(n),z_(n)))/(m_(1)+m_(2)+...+m_(n))`
`=(Sigmam_(i)(x_(i),y_(i),z_(i)))/(Sigmam_(i))` where `i=1,2,3,....,n`
OR
X-coordinate, Y-coordinate and Z-coordinate of centre of mass of a system,
`X=(m_(1)x_(1)+m_(2)x_(2)+...+m_(n)x_(n))/(m_(1)+m_(2)+...+m_(n))=(Sigmam_(i)x_(i))/(M)`
`Y=(m_(1)y_(1)+m_(2)y_(2)+...+m_(n)y_(n))/(m_(1)+m_(2)+...+m_(n))=(Sigmam_(i)y_(i))/(M)`
`Z=(m_(1)z_(1)+m_(2)z_(2)+...+m_(n)z_(n))/(m_(1)+m_(2)+...+m_(n))=(Sigmam_(i)z_(i))/(M)`
where `i=1,2,3,...,n` and
`m_(1)+m_(2)+.....+m_(n)=Sigmam_(i)=M` is the total mass of system.
If we want to find the position vector of centre of mass in i then position vector of `vecr=x_(i)hati+y_(i)hatj+z_(i)hatk` and the position vector of centre of mass
`vecR=(Xhati+Yhatj+Zhatk)`
`therefore vecR=(Sigmam_(i)vecr_(i))/(M)`
If the origin of frame of reference (coordinate system) taken as centre of mass of a system then for the system of particles `Sigmam_(i)vec(r_(i))=0`
