How do you convert spherical coordinates into Cartesian coordinates?

To convert a point from spherical coordinates to Cartesian coordinates, use equations x=ρsinφcosθ,y=ρsinφsinθ, and z=ρcosφ. To convert a point from Cartesian coordinates to spherical coordinates, use equations ρ2=x2+y2+z2,tanθ=yx, and φ=arccos(z√x2+y2+z2).

Can you take the dot product of spherical coordinates?

Radius r appears as a scalar multiple in spherical coordinates. A simple illustration: the dot product (a,b) of vector a with spherical coordinates (r,θ,ϕ)=(1,0,0) and vector b with spherical coordinates (r,θ,ϕ)=(1,0,φ0) is cos(φ0).

What is the Jacobian for spherical coordinates?

Our Jacobian is then the 3×3 determinant ∂(x,y,z)∂(r,θ,z) = |cos(θ)−rsin(θ)0sin(θ)rcos(θ)0001| = r, and our volume element is dV=dxdydz=rdrdθdz. Spherical Coordinates: A sphere is symmetric in all directions about its center, so it’s convenient to take the center of the sphere as the origin.

Why do we prefer spherical coordinate system?

Spherical coordinates determine the position of a point in three-dimensional space based on the distance ρ from the origin and two angles θ and ϕ. If one is familiar with polar coordinates, then the angle θ isn’t too difficult to understand as it is essentially the same as the angle θ from polar coordinates.

What is the dot product of the unit vector i and i?

Given that the vectors are all of length one, the dot products are i⋅i=j⋅j=k⋅k=1.

What is the relation between Cartesian coordinate and spherical coordinate?

In summary, the formulas for Cartesian coordinates in terms of spherical coordinates are x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ.

What is the relationship between Cartesian and polar coordinates?

If (x, y) be the cartesian co-ordinates of the point whose polar co-ordinates are (r, θ), then we have, x = r cos θ and y = r sin θ. or, (2x² + 2y² – ax)² = a² (x² + y²), which is the required cartesian form of the given polar form of equation.

How is Phi value calculated?

Phi is most often calculated using by taking the square root of 5 plus 1 and divided the sum by 2:

  1. √5 + 1.
  2. BC = √5.
  3. DE = 1.
  4. BE = DC = (√5-1)/2+1 = (√5+1)/2 = 1.618 … = Phi.
  5. BD = EC = (√5-1)/2 = 0.618… = phi.