Transform the vector A = 3i – 2j – 4k at P(2,3,3) to cylindrical coordinates

To transform the vector A = 3i – 2j – 4k at the point P(2,3,3) into cylindrical coordinates, we first need to express the Cartesian coordinates (x, y, z) in terms of cylindrical coordinates (r, θ, z):

– r = √(x² + y²)

– θ = arctan(y/x)

– z remains the same.

Given P(2, 3, 3):

– x = 2, y = 3, z = 3

– r = √(2² + 3²) = √(4 + 9) = √13

– θ = arctan(3/2)

Now we express the vector A in cylindrical coordinates:

– The unit vectors in cylindrical coordinates are ê_r, ê_θ, and ê_z. Here, ê_r points in the direction of the radius from the origin to the point (x, y) and ê_θ is perpendicular to ê_r.

To express A:
1. The z-component remains -4 (as z does not change).
2. The vector A can be decomposed into cylindrical components:

– A = A_r * ê_r + A_θ * ê_θ + A_z * ê_z

Calculating the components:

– A_r (the radial component) = (3i – 2j) • ê_r:

– To find ê_r:

– ê