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Chapter 1: Problem 2

If \(\vec{A}\) and \(\vec{B}\) are vectors and \(\vec{B}=-\vec{A},\) which of the following is true? a) The magnitude of \(\vec{B}\) is equal to the negative of the magnitude of \(\vec{A}\). b) \(\vec{A}\) and \(\vec{B}\) are perpendicular. c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\) d) \(\vec{A}+\vec{B}=2 \vec{A}\).

Short Answer

Expert verified
a) \(|\vec{B}| = 2|\vec{A}|\) b) \(\vec{A}\) is perpendicular to \(\vec{B}\) c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\) d) \(\vec{A} + \vec{B} = 2\vec{A}\) Answer: c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\).

Step by step solution

01

Determine the magnitudes

Since \(\vec{B} = -\vec{A}\), the magnitudes \(|\vec{B}|\) and \(|\vec{A}|\) must satisfy the relation \(|\vec{B}| = |-1||\vec{A}|\). Since the absolute value of \(-1\) is 1, the magnitudes must be equal, i.e., \(|\vec{B}| = |\vec{A}|\). Thus, the statement "a" is not correct.
02

Check for perpendicularity

To determine if \(\vec{A}\) and \(\vec{B}\) are perpendicular, we can calculate their dot product. Using the relation \(\vec{B} = -\vec{A}\), we get \(\vec{A}\cdot\vec{B} = \vec{A}\cdot(-\vec{A}) = -\vec{A}\cdot\vec{A} = -|\vec{A}|^2\). Now, if they were perpendicular, the dot product would equal \(0\). However, since both vectors have non-zero magnitudes, the dot product is non-zero, and statement "b" is incorrect.
03

Compare direction angles

Based on the given relation between vectors, \(\vec{B}\) has the same magnitude as \(\vec{A}\) but is pointing in the opposite direction. Therefore, their direction angles must differ by 180°, either adding or subtracting this value. Thus, statement "c" is correct.
04

Calculate vector sum

To verify whether statement "d" is correct, we first calculate the vector sum of both \(\vec{A}\) and \(\vec{B}\). Since \(\vec{B} = -\vec{A}\), we get \(\vec{A} + \vec{B} = \vec{A} + (-\vec{A}) = \vec{A} - \vec{A} = \overrightarrow{0}\). This result differs from statement "d", which states that their sum should be \(2\vec{A}\), so this statement is incorrect. From the analysis, the correct choice is "c". The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\).

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