Transform the vector B=yi+(x+z)j located at point (-2,6,3) into cylindrical coordinates.

To transform the vector ( mathbf{B} = y mathbf{i} + (x + z) mathbf{j} ) located at the point ((-2, 6, 3)) into cylindrical coordinates, we start by identifying the Cartesian coordinates and then convert them as follows:

1. The cylindrical coordinates ((r, theta, z)) are defined as:

– ( r = sqrt{x^2 + y^2} )

– ( theta = tan^{-1}left(frac{y}{x}right) )

– ( z ) remains the same in both cylindrical and Cartesian coordinates.

2. For the point ((-2, 6, 3)):

– ( x = -2 )

– ( y = 6 )

– ( z = 3 )

3. Calculate ( r ):

[

r = sqrt{(-2)^2 + 6^2} = sqrt{4 + 36} = sqrt{40} = 2sqrt{10}

]

4. Calculate ( theta ):

[

theta = tan^{-1}left(frac{6}{-2}right) = tan^{-1}(-3)

]

However, since the point is in the second quadrant (where ( x < 0 ) and ( y >