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Chapter 1: Problem 81

If \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{C}\), then (A) \(\vec{A}=\vec{C}\) always (B) \(\vec{A} \neq \vec{C}\) always (C) \(\vec{A}\) may not be equal to \(\vec{C}\) (D) None of these

Short Answer

Expert verified
(C) \(\vec{A}\) may not be equal to \(\vec{C}\)

Step by step solution

01

Write the dot product equations

We have the following equations: (1) \(\vec{A} \cdot \vec{B} = \|\vec{A}\| \|\vec{B}\| \cos\theta_{AB}\) (2) \(\vec{B} \cdot \vec{C} = \|\vec{B}\| \|\vec{C}\| \cos\theta_{BC}\) And we know that \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{C}\).
02

Compare the two dot products

As \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{C}\), we have: \(\|\vec{A}\| \|\vec{B}\| \cos\theta_{AB} = \|\vec{B}\| \|\vec{C}\| \cos\theta_{BC}\) If \(\vec{B}\) is the zero-vector, the equation above is trivially true and we cannot say anything about the relationship between \(\vec{A}\) and \(\vec{C}\). So, we assume that \(\vec{B}\) is non-zero. Divide both sides by \(\|\vec{B}\|\): \(\|\vec{A}\| \cos\theta_{AB} = \|\vec{C}\| \cos\theta_{BC}\)
03

Analyze the relationship between \(\vec{A}\) and \(\vec{C}\)

From the equation above, we cannot conclude that \(\|\vec{A}\| = \|\vec{C}\|\) or \(\theta_{AB} = \theta_{BC}\), so there are no guarantees that \(\vec{A}\) and \(\vec{C}\) are the same or that they are different. The only conclusion we can make is that the magnitudes of the projections of \(\vec{A}\) and \(\vec{C}\) onto \(\vec{B}\) are equal. Thus, the correct answer is: (C) \(\vec{A}\) may not be equal to \(\vec{C}\)

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