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Correct Answer: c is a vector.
the cross product of two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), is defined as \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \). the result of this operation, \( \mathbf{c} \), is always a vector. this is a fundamental property of the cross product in vector algebra. the magnitude of \( \mathbf{c} \) is given by \( ||\mathbf{a}|| \, ||\mathbf{b}|| \sin(\theta) \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \), and \( ||\mathbf{a}|| \) and \( ||\mathbf{b}|| \) are the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) respectively. the direction of \( \mathbf{c} \) is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \), following the right-hand rule.
it's important to note that \( \mathbf{c} \) is not necessarily a unit vector. the magnitude of \( \mathbf{c} \) depends on the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \), as well as the sine of the angle between them. therefore, \( \mathbf{c} \) will only be a unit vector if \( ||\mathbf{a}|| \, ||\mathbf{b}|| \sin(\theta) = 1 \), which is not generally the case for arbitrary vectors \( \mathbf{a} \) and \( \mathbf{b} \).
additionally, \( \mathbf{c} \) does not bisect the angle between \( \mathbf{a} \) and \( \mathbf{b} \). instead, as stated earlier, \( \mathbf{c} \) is perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \). the misconception that \( \mathbf{c} \) bisects the angle between \( \mathbf{a} \) and \( \mathbf{b} \) may arise from confusing the cross product with other vector operations, such as the dot product or vector addition.
concerning the magnitude of \( \mathbf{c} \) relative to the sum of \( \mathbf{a} \) and \( \mathbf{b} \), it is not accurate to claim that \( ||\mathbf{c}|| \) is always greater than \( ||\mathbf{a} + \mathbf{b}|| \). the actual magnitudes depend on the specific vectors and their relative directions. for example, if \( \mathbf{a} \) and \( \mathbf{b} \) are parallel (\( \theta = 0^\circ \) or \( \theta = 180^\circ \)), then \( \sin(\theta) = 0 \), making \( ||\mathbf{c}|| = 0 \), regardless of the magnitude of \( ||\mathbf{a} + \mathbf{b}|| \).
in conclusion, the only universally true statement about the vector \( \mathbf{c} \) resulting from the cross product \( \mathbf{a} \times \mathbf{b} \) is that \( \mathbf{c} \) is indeed a vector, perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \) and following the right-hand rule. the other properties, such as being a unit vector or bisecting the angle between \( \mathbf{a} \) and \( \mathbf{b} \), are not generally true and depend on the specific vectors involved.
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