(a) Explanation Given data: The vectors are given as, A = x ^ 2 − y ^ 3 + z ^ 1 (1) B = x ^ B x + y ^ 2 + z ^ B z (2) Calculation: The general form of a vector is given as, R = ( x ^ R x + y ^ R y + z ^ R z ) (3) Compare equation (1) and (3), the values are written as, R x = 2 R y = − 3 R z = 1 The magnitude of the vector R is calculated as, R = ( R x ) 2 + ( R y ) 2 + ( R z ) 2 Here, R is the magnitude of the vector R . Substitute 2 for R x , − 3 for R y , 1 for R z and A for R in the above expression. A = ( 2 ) 2 + ( − 3 ) 2 + ( 1 ) 2 = 4 + 9 + 1 = 14 The unit vector in the direction of A is given as, a ^ A = A A Here, a ^ A is the unit vector in the direction of A , A is the magnitude of vector A . Substitute ( x ^ 2 − y ^ 3 + z ^ 1 ) for A and 14 for A in the above expression. a ^ A = ( x ^ 2 − y ^ 3 + z ^ 1 ) 14 Compare equation (2) and (3), the values are written as, R x = B x R y = 2 R z = B z The magnitude of the vector R is calculated as, R = ( R x ) 2 + ( R y ) 2 + ( R z ) 2 Here, R is the magnitude of the vector R . Substitute B x for R x , 2 for R y , B z for R z and B for R in the above expression. B = ( B x ) 2 + ( 2 ) 2 + ( B z ) 2 = ( B x ) 2 + 4 + ( B z ) 2 The unit vector in the direction of B is given as, a ^ B = B B Here, a ^ B is the unit vector in the direction of B , B is the magnitude of vector B . Substitute x ^ B x + y ^ 2 + z ^ B z for B and ( B x ) 2 + 4 + ( B z ) 2 for B in the above expression. a ^ B = ( x ^ B x + y ^ 2 + z ^ B z ) ( B x ) 2 + 4 + ( B z ) 2 It is given that if A is parallel to B , then their directions are equal or opposite and it is written as, a ^ A = ± a ^ B Here, a ^ A is the direction of vector A , a ^ B is the direction of vector B (b) To determine A relation between B x and B z if A is perpendicular to B .

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Solenoidal Vector Field

In Physics and Mathematics vector calculus attached to each point in a subset of space, there is an assignment of a vector in a field called a vector field.

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Chapter 3, Problem 4P

(a)

To determine

The values of Bx and Bz if A is parallel to B.

(b)

To determine

A relation between Bx and Bz if A is perpendicular to B.

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Chapter 3 Solutions

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The values of B x and B z if A is parallel to B .

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The values of Bx and Bz if A is parallel to B.

A relation between Bx and Bz if A is perpendicular to B.

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Chapter 3 Solutions