Calculus 3 The 3-D Coordinate System

The 3D coordinate plane

Plotting points Plot the point (-4,3,-5) Plot the point (3,-2,-6)

All the points are in the form (3,y) All the points are in the form (3,y,z)

z=0 is the xy plane y=0 is the xz plane x=0 is the yz plane What surfaces in  3 are represented by the following equations? z = 3 y = 5

Describe and sketch the surface in  3 represented by the equation y = x.

Since we have not specified z at any value of z this equation must be satisfied. So, at any value z we have a circle of radius 2 centered on the z-axis. This means that we have a cylinder of radius 2 centered on the z-axis.

Distance Formula What is the distance between the point P (2,-1,7) and Q(1,-3,5)?

Equation of a Sphere We want to find an equation of a sphere with radius r and center C(h, k, l). The point P(x, y, z) is on the sphere if and only if |PC | = r. Squaring both sides gives |PC | 2 = r 2. (x – h) 2 + (y – k) 2 + (z – l) 2 = r 2

Find the equation of the sphere whose center is (2,-4,3) and whose radius is 3? (x – h) 2 + (y – k) 2 + (z – l) 2 = r 2 (x – 2) 2 + (y + 4) 2 + (z – 3) 2 = 3 2 (x – 2) 2 + (y + 4) 2 + (z – 3) 2 = 9

Show that x 2 + y 2 + z 2 + 4x – 6y + 2z + 6 = 0 is a circle, and find its center and radius. We are going to use completing the square. (when the a term is 1, to find the c term we square b/2) x 2 + 4x +___ + y 2 – 6y+ ____ + z 2 + 2z + ____ = -6 x 2 + 4x y 2 – 6y+ 9 + z 2 + 2z + 1 = (x+2) 2 + (y-3) 2 + (z+1) 2 =8 Therefore, it is a circle with center (-2,3,-1) and radius

Home work P 641 3,5,7,11,13,15b,17