Universal Law of Gravitation

The figure shows a three-dimensional coordinate system with axes labelled X, Y, and Z. The origin is marked O. Two point masses, $m_1$ and $m_2$, are placed at different locations in this space. From the origin, a position vector $\mathbf{r}_1$ points to $m_1$, and a position vector $\mathbf{r}_2$ points to $m_2$. A third vector, labelled $\mathbf{r}$, is drawn from $m_1$ to $m_2$. The caption tells you that $\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1$, and that the gravitational force on $m_1$ due to $m_2$ acts along this vector $\mathbf{r}$.

The physical idea is simple but profound: gravity is a force that acts along the straight line connecting two masses. The figure makes this directional dependence explicit. The vector $\mathbf{r}$ is not just any line — it is the displacement from $m_1$ to $m_2$. Its magnitude $|\mathbf{r}|$ is the distance between the two masses, and its direction points from $m_1$ toward $m_2$.

The textbook uses this figure to introduce the universal law of gravitation. The force on $m_1$ due to $m_2$ is written as

where $G$ is the universal gravitational constant, $r = |\mathbf{r}_2 - \mathbf{r}_1|$ is the distance between the masses, and $\hat{\mathbf{r}}$ is a unit vector pointing from $m_1$ to $m_2$. The minus sign tells you that the force on $m_1$ is directed opposite to $\hat{\mathbf{r}}$ — that is, toward $m_2$. In other words, the force is attractive.

The same figure also sets up the force on $m_2$ due to $m_1$, which is equal in magnitude and opposite in direction: …

The figure shows a single point mass $m_1$ at the centre, with three arrows pointing outward toward three other point masses: $m_2$ in the upper right, $m_3$ on the left, and $m_4$ below. Each arrow is labelled with the corresponding gravitational force vector: $\mathbf{F}_{12}$ (from $m_2$ on $m_1$), $\mathbf{F}_{13}$ (from $m_3$), and $\mathbf{F}_{14}$ (from $m_4$). The arrows are drawn in indigo, radiating from $m_1$ outward toward each other mass — this is the key visual: the forces act on $m_1$, but they point toward the masses that exert them. That is, $\mathbf{F}_{12}$ points from $m_1$ toward $m_2$, not away from $m_2$.

The physical idea is straightforward but foundational: gravity is a mutual attraction between every pair of masses. For a single mass $m_1$ surrounded by several others, the net gravitational force it experiences is the vector sum of the individual forces from each other mass. The figure makes clear that you cannot simply add magnitudes — direction matters. Each force $\mathbf{F}_{1j}$ acts along the line joining $m_1$ and $m_j$, and its magnitude is given by the universal law of gravitation:

where $G = 6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2$ is the universal gravitational constant, and $r_{12}$ is the distance between $m_1$ and $m_2$. The same form holds for $\mathbf{F}_{13}$ and $\mathbf{F}_{14}$, with $r_{13}$ and $r_{14}$ respectively.

The net force on $m_1$ is then:

Each term is a vector: $\mathbf{F}_{1j} = -G \frac{m_1 m_j}{r_{1j}^2} \,\hat{\mathbf{r}}_{1j}$, where $\hat{\mathbf{r}}_{1j}$ is a unit vector pointing from $m_1$ to $m_j$. The negative sign (often omitted in magnitude form) reminds you that the force on $m_1$ is attractive — it points toward $m_j$, opposite to the direction of $\hat{\mathbf{r}}_{1j}$ if you define $\hat{\mathbf{r}}_{1j}$ as pointing from $m_1$ to $m_j$.