Prove the product rule for the dot product of vector functions: d/dt(U · V) = U · dV/dt + dU/dt · V

The vectors $\vec{U}$ and $\vec{V}$ are given as:

$$\vec{U} = F_1(t)\vec{i} + g_1(t)\vec{j} + h_1(t)\vec{k}$$ $$\vec{V} = F_2(t)\vec{i} + g_2(t)\vec{j} + h_2(t)\vec{k}$$

The dot product of two vectors is the sum of the products of their corresponding components:

$$\vec{U} \cdot \vec{V} = F_1(t)F_2(t) + g_1(t)g_2(t) + h_1(t)h_2(t)$$

We apply the product rule for scalar functions to each term:

$$\frac{d}{dt}(\vec{U} \cdot \vec{V}) = \frac{d}{dt}(F_1(t)F_2(t)) + \frac{d}{dt}(g_1(t)g_2(t)) + \frac{d}{dt}(h_1(t)h_2(t))$$ $$\frac{d}{dt}(\vec{U} \cdot \vec{V}) = (F_1'(t)F_2(t) + F_1(t)F_2'(t)) + (g_1'(t)g_2(t) + g_1(t)g_2'(t)) + (h_1'(t)h_2(t) + h_1(t)h_2'(t))$$

Rearranging the terms, we get:

$$\frac{d}{dt}(\vec{U} \cdot \vec{V}) = (F_1(t)F_2'(t) + g_1(t)g_2'(t) + h_1(t)h_2'(t)) + (F_1'(t)F_2(t) + g_1'(t)g_2(t) + h_1'(t)h_2(t)) \quad (1)$$

$$\frac{d\vec{U}}{dt} = F_1'(t)\vec{i} + g_1'(t)\vec{j} + h_1'(t)\vec{k}$$ $$\frac{d\vec{V}}{dt} = F_2'(t)\vec{i} + g_2'(t)\vec{j} + h_2'(t)\vec{k}$$

$\vec{U} \cdot \frac{d\vec{V}}{dt} + \frac{d\vec{U}}{dt} \cdot \vec{V}$.
First, calculate $\vec{U} \cdot \frac{d\vec{V}}{dt}$:

$$\vec{U} \cdot \frac{d\vec{V}}{dt} = (F_1(t)\vec{i} + g_1(t)\vec{j} + h_1(t)\vec{k}) \cdot (F_2'(t)\vec{i} + g_2'(t)\vec{j} + h_2'(t)\vec{k})$$ $$\vec{U} \cdot \frac{d\vec{V}}{dt} = F_1(t)F_2'(t) + g_1(t)g_2'(t) + h_1(t)h_2'(t)$$

Next, calculate $\frac{d\vec{U}}{dt} \cdot \vec{V}$:

$$\frac{d\vec{U}}{dt} \cdot \vec{V} = (F_1'(t)\vec{i} + g_1'(t)\vec{j} + h_1'(t)\vec{k}) \cdot (F_2(t)\vec{i} + g_2(t)\vec{j} + h_2(t)\vec{k})$$ $$\frac{d\vec{U}}{dt} \cdot \vec{V} = F_1'(t)F_2(t) + g_1'(t)g_2(t) + h_1'(t)h_2(t)$$

Now, sum these two results:

$$\vec{U} \cdot \frac{d\vec{V}}{dt} + \frac{d\vec{U}}{dt} \cdot \vec{V} = (F_1(t)F_2'(t) + g_1(t)g_2'(t) + h_1(t)h_2'(t)) + (F_1'(t)F_2(t) + g_1'(t)g_2(t) + h_1'(t)h_2(t)) \quad (2)$$

Comparing equation (1) and equation (2), we see that both sides are identical.

$$\boxed{\frac{d}{dt}(\vec{U} \cdot \vec{V}) = \vec{U} \cdot \frac{d\vec{V}}{dt} + \frac{d\vec{U}}{dt} \cdot \vec{V}}$$

This shows the product rule for the dot product of vector functions.