Chapter 12 – Vectors and the Geometry of Space
13
Chapter 12 – Vectors and the Geometry of Space 12.1 – Three Dimensional Coordinate Systems 12.1 – Three Dimensional Coordinate Systems 1
-
Upload
faith-hull -
Category
Documents
-
view
48 -
download
2
description
Chapter 12 – Vectors and the Geometry of Space. 12.1 – Three Dimensional Coordinate Systems. 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. 3D Coordinate System. The purple plane is the yz -plane. - PowerPoint PPT Presentation
Transcript of Chapter 12 – Vectors and the Geometry of Space
1
Chapter 12 – Vectors and the Geometry of Space12.1 – Three Dimensional Coordinate Systems
12.1 – Three Dimensional Coordinate Systems
12.1 – Three Dimensional Coordinate Systems
2
Coordinate axesThe 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 Systems
3
3D Coordinate SystemThe purple plane
is the yz-plane.The pink plane is
the xz-plane.The green plane
is the xy-plane.
12.1 – Three Dimensional Coordinate Systems
4
Points in 3D Coordinate SystemsOrdered 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.
12.1 – Three Dimensional Coordinate Systems
5
3D Coordinate SystemThe 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.
12.1 – Three Dimensional Coordinate Systems
6
DefinitionThe 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.
( , , ) | , ,x y z x y z
3
3
12.1 – Three Dimensional Coordinate Systems
7
DefinitionDistance Formula in 3D: the distance
|P1P2|between the points P1(x1,y1,z1) and P2(x2,y2,z2) is
2 2 2
1 2 2 1 2 1 2 1PP x x y y z z
12.1 – Three Dimensional Coordinate Systems
8
DefinitionEquation 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
2 2 22r x h y k z l
2 2 2 2r x y z
12.1 – Three Dimensional Coordinate Systems
9
Example 1 – Page 790 #4What 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.
12.1 – Three Dimensional Coordinate Systems
10
Example 2 – Page 790 # 14Find an equation of the sphere
that passes through the origin and whose center is (1,2,3).
12.1 – Three Dimensional Coordinate Systems
11
Example 3 – Page 790 #16Show that the equation
represents a sphere and find its center and radius.2 2 2 8 6 2 17 0x y z x y z
12.1 – Three Dimensional Coordinate Systems
12
Example 4 – Page 791 #20Find an equation of a sphere if
one of its diameters has endpoints (2,1,4) and (4,3,10).
12.1 – Three Dimensional Coordinate Systems
13
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 3◦Example 5
