To clarify, I assume you're referring to the expression:
(G * m * m / r^3) * (unit vector of r)
where G is the gravitational constant, m is the mass, r is the distance, and the unit vector of r represents the direction of the vector r.
To prove that this expression is equal to (G * m * m / r^2), we can employ a mathematical technique called vector calculus.
Let's start by expressing the unit vector of r as r/r, where r is the magnitude of the vector r. Therefore, we can rewrite the expression as:
(G * m * m / r^3) * (r/r)
Now, we can simplify this expression by canceling out one of the r terms:
(G * m * m / r^2) * (1/r)
Now, we have (1/r) remaining, which can be further simplified as r^-1. By applying the laws of exponents, we can rewrite it as:
(G * m * m / r^2) * (r^-1)
Now, using the properties of multiplication, we can combine the two terms:
(G * m * m * r^-1) / r^2
Finally, applying the laws of exponents again, we know that r^-1 can be expressed as 1/r, so the expression becomes:
(G * m * m / r^2) * (1/r)
We can see that this is the same expression we started with, which proves that:
(G * m * m / r^3) * (unit vector of r) = (G * m * m / r^2)
Thus, the equation is mathematically valid.