Suppose every particle of a body is positioned at a distance R from...

Let the origin be $O(0,0,0)$. The position vector of the $i$-th particle of mass $m_i$ is $\vec{r}_i$.

Given that every particle is at a distance $R$ from the origin, we have $|\vec{r}_i| = R$ for all $i$.

The position vector of the centre of mass is given by:

$\vec{r}_{cm} = \dfrac{\sum m_i \vec{r}_i}{\sum m_i}$

Taking the magnitude on both sides, we get:

$|\vec{r}_{cm}| = \left| \dfrac{\sum m_i \vec{r}_i}{\sum m_i} \right|$

Using the triangle inequality for vectors, \left| \sum \vec{A}_i \right| \leq \sum |\vec{A}_i|$, we have:<br><br>$|\vec{r}_{cm}| \leq \dfrac{\sum m_i |\vec{r}_i|}{\sum m_i}$<br><br>Substituting$|\vec{r}_i| = R$:<br><br>$|\vec{r}_{cm}| \leq \dfrac{\sum m_i R}{\sum m_i}$<br><br>$|\vec{r}_{cm}| \leq R \dfrac{\sum m_i}{\sum m_i}$<br><br>$|\vec{r}_{cm}| \leq R$<br><br>Therefore, the distance of the centre of mass from the origin is$\leq R$.$