This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hey Ðƴŋâmïč̣, good to see you again. To show the product rule for the dot product of vector functions, we will start by defining the given vectors and then compute both sides of the identity (d)/(dt)(U · V) = U · dVdt + dUdt · V. Step 1: Define the given vectors. The vectors U and V are given as: U = F_1(t)i + g_1(t)j + h_1(t)k V = F_2(t)i + g_2(t)j + h_2(t)k Step 2: Calculate the dot product U · V. The dot product of two vectors is the sum of the products of their corresponding components: U · V = F_1(t)F_2(t) + g_1(t)g_2(t) + h_1(t)h_2(t) Step 3: Differentiate U · V with respect to t. We apply the product rule for scalar functions to each term: (d)/(dt)(U · V) = (d)/(dt)(F_1(t)F_2(t)) + (d)/(dt)(g_1(t)g_2(t)) + (d)/(dt)(h_1(t)h_2(t)) (d)/(dt)(U · 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: (d)/(dt)(U · 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)) (1) Step 4: Calculate the derivatives of the individual vectors, dUdt and dVdt. dUdt = F_1'(t)i + g_1'(t)j + h_1'(t)k dVdt = F_2'(t)i + g_2'(t)j + h_2'(t)k Step 5: Calculate the right-hand side of the identity: U · dVdt + dUdt · V. First, calculate U · dVdt: U · dVdt = (F_1(t)i + g_1(t)j + h_1(t)k) · (F_2'(t)i + g_2'(t)j + h_2'(t)k) U · dVdt = F_1(t)F_2'(t) + g_1(t)g_2'(t) + h_1(t)h_2'(t) Next, calculate dUdt · V: dUdt · V = (F_1'(t)i + g_1'(t)j + h_1'(t)k) · (F_2(t)i + g_2(t)j + h_2(t)k) dUdt · V = F_1'(t)F_2(t) + g_1'(t)g_2(t) + h_1'(t)h_2(t) Now, sum these two results: U · dVdt + dUdt · 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)) (2) Step 6: Compare the results from Step 3 and Step 5. Comparing equation (1) and equation (2), we see that both sides are identical. (d)/(dt)(U · V) = U · dVdt + dUdt · V This shows the product rule for the dot product of vector functions. Drop the next question!