# How do you convert spherical coordinates into Cartesian coordinates?

Table of Contents

## 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:

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