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MTEL Physics (11) Practice Tests & Test Prep by Exam Edge

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MTEL 11 Practice Test Features

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MTEL Physics Sample Test

1 of 5

The cross product of two vectors is a vector aligned in a direction perpendicular to the two vectors which completes a right-handed system. Why is the right-handedness constraint necessary?

Correct Answer:

there exist two vectors with the same magnitude that are perpendicular to both a and b

the cross product of two vectors, denoted as **a** × **b**, results in a third vector that is perpendicular to the plane containing **a** and **b**. mathematically, this vector can be expressed as ||**a**|| ||**b**|| sin(θ) **n**, where θ is the angle between **a** and **b**, and **n** is a unit vector perpendicular to both **a** and **b**. the sine function indicates that the magnitude of the cross product depends on the sine of the angle between the two vectors, which reaches its maximum value when the vectors are orthogonal.

the right-handedness constraint specifies the direction of the vector **n**. in a right-handed coordinate system, if you curl the fingers of your right hand from the first vector **a** towards the second vector **b**, your thumb points in the direction of **n**. this convention is essential because, for any two non-collinear vectors **a** and **b**, there exist exactly two unit vectors that are perpendicular to both: **n** and -**n**. without a consistent rule like the right-handed system, the direction of the cross product would be ambiguous.

the choice of the right-handed system over a left-handed one is arbitrary but universally adopted for consistency across mathematical and physical applications. this consistency is crucial in fields like physics and engineering, where cross products are used to determine directions of forces, magnetic fields, and rotations. adopting a standard convention allows everyone in the scientific community to communicate and understand results unambiguously.

in summary, the right-handedness constraint in the definition of the cross product is necessary to eliminate ambiguity in the direction of the resulting vector. it ensures a consistent and universally accepted method for calculating and representing vector directions in three-dimensional space, facilitating clear and effective communication in scientific and mathematical contexts.