The integral $\int \vec{r} dm = 0$ for a homogeneous body is defined for which point?

$A$ uniform wire is bent into a $U$ shape with side lengths $l, 2l,$ and $l$. The coordinates of the center of mass of each segment are shown in the figure. The coordinates of the center of mass of the system are:

Three identical spheres,each of mass $1 \ kg$,are arranged as shown in the figure. If their centers of mass are at $P$,$Q$,and $R$ respectively,what is the distance of the center of mass of this system from point $P$?

$A$ rigid body consists of a $3 \ kg$ mass and a $2 \ kg$ mass connected by a massless rod. The $3 \ kg$ mass is at $\vec{r}_1 = (2 \hat{i} + 5 \hat{j}) \ m$ and the $2 \ kg$ mass is at $\vec{r}_2 = (4 \hat{i} + 2 \hat{j}) \ m$. Find the length of the rod and the coordinates of the center of mass.

Three identical uniform thin rods each of mass $m$ and length $L$ are arranged in the $XY$ plane as shown in the figure. $A$ fourth uniform thin rod of mass $3m$ is placed as shown in the figure in the $XY$ plane. The value of length of the fourth rod such that the centre of mass of all the four rods lies at the origin is

The position vectors of two particles of masses $1 \ kg$ and $3 \ kg$ are $\hat{i} + 2\hat{j} + \hat{k}$ and $-3\hat{i} - 2\hat{j} + \hat{k}$ respectively. The position vector of the center of mass of this system is: