How do I express Z and R unit vectors in different coordinate systems?

[zekester]

at point T(2,3,-4) in rectangular coordinate system, how would I express the Z unit vector in the spherical system and the R unit vector in the rectangular system? I know T in spherical coordinates is (5.385,-42 degrees,56.3 degrees) but i have no idea how i would express a unit vector in a different coordinate system.

 

[andrevdh]

The x unit vector is the vector (1,0,0) in the rectangular coordinate system. So all you need to do is transform the coordinates from one system to the other.

 

[quicknote]

To express the Z unit vector in spherical coordinates, we first need to understand that the Z unit vector represents the direction of the Z-axis in the rectangular coordinate system. In spherical coordinates, this would be equivalent to the direction of the polar axis, which is represented by the angle θ. In this case, the Z unit vector would be represented by (sinθ, 0, cosθ) or (0, 0, cosθ) since the point is on the equatorial plane (θ = 90 degrees). Substituting the values from the given point T, we get (0, 0, -0.8) as the Z unit vector in spherical coordinates.

To express the R unit vector in rectangular coordinates, we first need to understand that the R unit vector represents the direction of the radius vector, which is the distance from the origin to the point T. In spherical coordinates, this would be equivalent to the direction of the radial distance, which is represented by the angle φ. In this case, the R unit vector would be represented by (cosφsinθ, sinφsinθ, cosθ) or (2/√29, 3/√29, -4/√29) since the point T is located at (2,3,-4). Therefore, the R unit vector in rectangular coordinates would be (2/√29, 3/√29, -4/√29).