Write the vector V=i+2j+3k at the point (x,y,z)=(1,1,0)

[jlmac2001]

jlmac2001 Question:

Write the vector V=i+2j+3k at the point (x,y,z)=(1,1,0) in terms of the spherical polar coordinate unit vectors r, theta and phi at that point. Do this again at the point (x,y,z)=(1,1,1). Check your answers by checking the norm of both vectors and comparing to the norm in Cartesian coordinates and by making sure that the signs of each component are as you would expect.



How would I answer this in terms of spherical polar coordinate unit vectors r, theta and phi?

 

[arildno]

What are the transformation rules between the unit vectors in Cartesian and sph. coordinates?
That's all you need to take into account

 

[M98Ranger]

To write the vector V=i+2j+3k at the point (x,y,z)=(1,1,0) in terms of spherical polar coordinate unit vectors, we need to first convert the vector components from Cartesian coordinates to spherical coordinates. This can be done using the following equations:

r = √(x^2 + y^2 + z^2)

θ = arccos(z/r)

φ = arctan(y/x)

Substituting the given values (x,y,z)=(1,1,0) into these equations, we get:

r = √(1^2 + 1^2 + 0^2) = √2

θ = arccos(0/√2) = π/2

φ = arctan(1/1) = π/4

Therefore, the vector V in terms of spherical polar coordinate unit vectors at the point (x,y,z)=(1,1,0) is:

V = (i cos φ sin θ + j sin φ sin θ + k cos θ) r

Substituting the values of φ and θ, we get:

V = (√2/2 i + √2/2 j + 0k) √2

= i + j

Similarly, at the point (x,y,z)=(1,1,1), the vector V in terms of spherical polar coordinate unit vectors would be:

r = √(1^2 + 1^2 + 1^2) = √3

θ = arccos(1/√3) = π/3

φ = arctan(1/1) = π/4

Therefore, V = (i cos φ sin θ + j sin φ sin θ + k cos θ) r

= (√3/2 i + √3/2 j + 1/√3 k) √3

= √3 i + √3 j + k

To check our answers, we can calculate the norm of both vectors in Cartesian coordinates and compare it to the norm in spherical coordinates. The norm of V in Cartesian coordinates is √(1^2 + 2^2 + 3^2) = √14. And in spherical coordinates, the norm is r = √3 for the point (1,1,1)