Problem 1

Describe the set of all points in that satisfy

(a).

All the points on the surface of a sphere of radius 3 centred at .

(b).

All the points on the interior of a sphere of radius 3 centred at .

Problem 2

Describe and sketch all points in that satisfy

(a).

All points on the 45 degree line between the x and y axes

(b).

All points that cross the y-axis at and form a 45º angle x-axis and the negative part of the y-axis

(c).

All points on the circle of radius 2 centred at the origin.

(d).

Points on the circle of radius 1 centred around .

(e).

Points within the circle of radius 1 centred around .

Problem 3

Describe all points in that satisfy the following conditions.

(a).

The plane which contains the y-axis and forms a 45º angle with the xy-plane and the yz-plane.

(b).

The plane through , and .

(c).

The points on the sphere of radius 2 centred around the origin.

(d). ,

The points on the circle of radius centred around the point in the plane .

(e).

The cylinder centred on the z-axis of radius 2 which extends into the z direction.

(f).

A paraboloid from the origin extending upwards consisting of a vertical stack of horizontal circles. The intersection of the surface with the yz-plane is the parabola .

Problem 4

Let be the point .

(a). Find the Distance from to the -plane

(b). Find the Distance from to the -plane

(c). Find the Distance from to the point on the -axis

(d). Find the point on the -axis that is Closest to

The closest point on the -axis is the point .

(e). What is the Distance from to the -axis

The distance from to the -axis is

Problem 5

Consider any triangle. Pick a coordinate system so that one vertex is at the origin and a second vertex is on the positive –axis. Call the coordinates of the second vertex and those of the third vertex . Find the circumscribing circle (the circle that goes through all three vertices).

Problem 6

A certain surface consists of all points such that the distance from to the point is equal to the distance from to the plane . Find an equation for the surface, sketch and describe it verbally.

The surface is a paraboloid centred at the origin where the intersection with the yz-plane forms the parabola .

Problem 7

Show that the set of all points that are as twice as far from as from is a sphere. Find its centre and radius.

The centre is at the point and it has a radius of $\ds \sqrt{\frac{265}{2