Chapter 12 – Vectors and the Geometry of Space 12.1 – Three Dimensional Coordinate Systems 1

Coordinate axes The 3D coordinate plane is used to represent any point in space. O is the origin. x, y, and z axis are all perpendicular 12.1 – Three Dimensional Coordinate Systems2

3D Coordinate System The purple plane is the yz-plane. The pink plane is the xz-plane. The green plane is the xy-plane – Three Dimensional Coordinate Systems3

Points in 3D Coordinate Systems Ordered pairs are in the form (x, y, z) called an ordered triple. Here we plotted point P by moving a units along the x-axis, b units along the y-axis, and c units along the z-axis – Three Dimensional Coordinate Systems4

3D Coordinate System The point P(a,b,c) determines a rectangular box. We drop a perpendicular from P to the xy-plane and get point Q called the projection of P on the xy-plane. Similarly, points P and S are the projections of P on the yz-plane and xz-plane respectively – Three Dimensional Coordinate Systems5

Definition The Cartesian product is the set of all ordered triples of real numbers and is denoted by. We have a one-to-one correspondence between the points P in space and the ordered triples (a,b,c) in. It is called a three dimensional rectangular coordinate system – Three Dimensional Coordinate Systems6

Definition Distance Formula in 3D: the distance |P 1 P 2 |between the points P 1 (x 1,y 1,z 1 ) and P 2 (x 2,y 2,z 2 ) is 12.1 – Three Dimensional Coordinate Systems7

Definition Equation of a Sphere – An equation of a sphere with center C(h,k,l) and radius r is In particular, if the center is the origin O, the equation of the sphere is 12.1 – Three Dimensional Coordinate Systems8

Example 1 – Page 790 #4 What are the projections of the point (2,3,5) on the xy-, yz-, and xz-planes? Draw a rectangular box with the origin and (2,3,5) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box – Three Dimensional Coordinate Systems9

Example 2 – Page 790 # 14 Find an equation of the sphere that passes through the origin and whose center is (1,2,3) – Three Dimensional Coordinate Systems10

Example 3 – Page 790 #16 Show that the equation represents a sphere and find its center and radius – Three Dimensional Coordinate Systems11

Example 4 – Page 791 #20 Find an equation of a sphere if one of its diameters has endpoints (2,1,4) and (4,3,10) – Three Dimensional Coordinate Systems12

More Examples The video examples below are from section 12.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 3 Example 3 ◦ Example 5 Example – Three Dimensional Coordinate Systems13