Final answer:
A paragraph proof utilizing the commutative property of addition confirms that m_C equals m_A by showing that the order of terms in the given equation m_A + m_B = m_B + m_C does not affect the outcome, allowing m_B to be canceled from both sides.
Step-by-step explanation:
To prove that m_C = m_A using the given m_A + m_B = m_B + m_C and the information provided, we should recall some basic properties of addition, specifically the commutative property which states that changing the order of addition does not change the sum (A + B = B + A). This property also applies to vectors, as demonstrated in various exercises which show that vectors added in any order yield the same sum. Applying this to our equation, since m_B appears on both sides, we can cancel it out, leading to m_A = m_C.
Here's the paragraph proof: Starting with the equation m_A + m_B and knowing the commutative property holds (A + B = B + A), we can assert that m_A + m_B = m_B + m_A. Given the provided relation that m_A + m_B = m_B + m_C, by the transitive property of equality, it follows that m_B + m_A = m_B + m_C. Canceling m_B from both sides of the equation gives us the desired result, m_A = m_C. This proves that the magnitudes of m_C and m_A are equal, derived from the commutative property of addition.