Physics for Scientists and Engineers, Vol. 1
Physics for Scientists and Engineers, Vol. 1
6th Edition
ISBN: 9781429201322
Author: Paul A. Tipler, Gene Mosca
Publisher: Macmillan Higher Education
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Chapter 10, Problem 2P
To determine

To Choose: The correct option.

Expert Solution & Answer
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Answer to Problem 2P

Option (b)

Explanation of Solution

Given:

Two nonzero vectors A and B .

Formula used:

The formula of vector product:

  A×B=|A||B|sinθ n^

·|A|= Magnitude of vector B

·|A|= Magnitude of vector A

·θ= Angle between the vectors A and B .

·n^= Unit vector perpendicular to the plane A×B .

Calculation:

When A and B are parallel:

  θ=0o

  A×B=|A||B|sinθ n^=|A||B|sin0o n^=|A||B|×0 n^                 (sin0o=0)=0 n^

The magnitude of the vector product:

  |A×B|=02=0

When A and B are perpendicular:

  θ=90o

  A×B=|A||B|sinθ n^=|A||B|sin90o n^=|A||B|×1 n^                 (sin 90o=1)=|A||B| n^

The magnitude of the vector product:

  |A×B|= ( coefficient of vector )2= ( | A || B | )2=|A||B|

When A and B are antiparallel:

  θ=180o

  A×B=|A||B|sinθ n^=|A||B|sin180o n^=|A||B|×0 n^                 (sin 180o=0)=0 n^

The magnitude of the vector product:

  |A×B|=02=0

When A and B are at 45o :

  θ=45o

  A×B=|A||B|sinθ n^=|A||B|sin45o n^=|A||B|×12 n^                 (sin 90o=1)=| A || B |2 n^

The magnitude of the vector product:

  |A×B|= ( coefficient of vector )2= ( | A || B | 2 )2=| A || B |2

The greatest magnitude of the vector product is |A||B| .

Conclusion:

Hence, when the vectors A and B are perpendicular to each other, the magnitude of the vector productwould be greatest.

Option (b) is correct.

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Physics for Scientists and Engineers, Vol. 1

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