Change from rectangular to spherical coordinates. (0,-2,0) ...

Key Concepts

Coordinate Transformation

Coordinate transformation is the process of converting the representation of a point from one coordinate system to another. This approach is vital in fields like physics, engineering, and computer graphics where the use of different coordinate systems can simplify mathematical expressions or better describe the geometry of a problem.

Conversion Formulas

Conversion from rectangular to spherical coordinates is achieved through specific formulas: the radial distance ? is calculated as the square root of the sum of the squares of the rectangular coordinates; the azimuthal angle ? is determined using the arctan2 function to properly account for the quadrant in the xy-plane; the polar angle ? is computed using the inverse cosine of the ratio of the z-coordinate to ?. These formulas ensure an accurate translation between coordinate systems.

Rectangular Coordinates

Rectangular coordinates represent points in space using three perpendicular axes (usually labeled x, y, and z). This system describes a point by its horizontal and vertical distances from an origin and is widely used in algebra and physics for its simplicity in representing linear geometry.

Spherical Coordinates

Spherical coordinates represent spatial points using three values: the radial distance from the origin (?), the azimuthal angle in the xy-plane from the positive x-axis (?), and the polar (or inclination) angle from the positive z-axis (?). This system is particularly useful in scenarios with spherical symmetry and simplifies the equations in problems involving rotations and radial distances.

Transcript

00:01 This question asks us to transform these rectangular coordinates into spherical coordinates.

00:05 To do that, we need to know a few equations.

00:09 Our first equation is that x is equal to row sine phi, cosine theta, y is equal to row sine phi, sine theta, z is equal to row cosine phi, and row squared is equal to x squared plus y squared plus z squared.

00:38 Since we know what x, y, and z are, we're able to start with the row squared equation, and we're able to plug that in to solve for row.

00:46 So let's do that.

00:47 We get that row squared is equal to zero squared plus negative two squared plus z squared plus z squared plus z squared.

00:54 0 squared.

00:56 And if we solve for row, we get that row squared is equal to 4, which just means that row is equal to 2.

01:04 Great.

01:05 Now we can plug it into the z equation because now all that's missing is our 5.

01:10 So z is 0, which is equal to 2, cosine of phi.

01:16 Now we must remember that phi is in between 0 and pi.

01:22 So we need to find where cosine is 0.

01:25 What you can plug in to make cosine 0 between 0 and pi.

01:30 The one number that satisfies this is when phi is equal to pi over 2.

01:38 So that means phi is pi over 2.

01:40 So then we can take all this and we either plug it into our x or y equation.

01:45 But we have to remember that whichever one we choose, we must make sure that it is true for the other.

01:51 So let's say if i were to choose x, i would need to make sure to plug it back in and make sure it was true for the y.

01:55 But on this case i'm going to choose y.

01:59 And so my y is negative 2 is equal to 2, sine of pi over 2, sine of theta.

02:09 So now i'm just going to solve for my theta.

02:13 So to start, i'm going to divide both sides by 2, so we get negative 1 is equal, and sine of pi over 2 is just 1.

02:20 So it's equal to 1, sine theta.

02:25 So when is sine of theta equal to negative 1? well, we know that theta is between 0 and 2 pi...

Please give Ace some feedback