Question

In: Math

1.2 A large circle on a sphere is a circle that forms the intersection of the...

1.2

A large circle on a sphere is a circle that forms the intersection of the sphere with a plane through the center of the sphere. Consider the large circle C that arises the intersection sphere x2 + y2 + z2 = 1 and the plane x + y + z = 0.

(a) Express the equations of the specified large circle C using spherical coordinates.

(b) Express the equations of the large circle C using cylindrical coordinates.

(c) Determine a parameterization of C by writing

r (t) = u cos t + v sin t,

where u and v are two orthogonal unit vectors in the plane that cut out C.

(d) Determine the speed and velocity of a particle traveling along the large circle

according to the parameterization in part (c), when the parameter refers to the time.

Solutions

Expert Solution

Consider the sphere

                                              ..... (1)

Also consider the plane

                                                    ..... (2)

(a)

The objective is to find the large circle which is the intersection of the sphere ad the plane.

Rewrite the plane (2) as

  

     

Substitute in (1).

                                            ..... (3)

This is the equation of the larger circle in certician form.

------------------------------------------------------------------------------------------------------

In spherical form the coordiates are

where is the radius

Substitute , and in (3)

This is the equation of the circle in spherical form.

-----------------------------------------------------------------------------------------------------------

(b)

The objective is to write the equation of the circle in cylindrical form.

In cylinndrical coordinate form t

Substitute , and in (3)

This is the equation of the circle in cylindrical form.


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