# Ch12Notes - MATH 214: INTRO TO VECTORS 1. The Geometry of 3...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 214: INTRO TO VECTORS 1. The Geometry of 3 D-Space We use the notation R 3 for 3 dimensional space with the standard x,y,z coordinates for points. Notice that fixing one of the coordinates gives a plane, fixing two of the coordinates gives a line and fixing three of the coordinates gives a point. Example 1. The xy-plane embeds in R 3 as the plane z = 0 . Example 2. The z-axis can be described by x = 0 = y . Three planes that appear particularly often are the coordinate planes z = 0 (sometimes called the xy-plane), x = 0 (sometimes called the yz-plane) and y = 0 (the xz-plane). The three coordinate planes divide R 3 into 8 pieces, and these pieces are sometimes called the octants of R 3 . We specify one octant in particular: Definition. The first octant of R 3 is the region given by { ( x,y,z ) | x,y,z ≥ } . Note that none of the other octants are given names. Example 3. The following shapes will be shown in class: (1) z ≥ (2) x =- 3 (3) z = 0 , x ≤ , y ≥ (4)- 1 ≤ y ≤ 1 (5) y =- 2 ,z = 2 Given any graph in the xy-plane, there are many ways that we can view it as sitting inside R 3 (we say that we embed it in R 3 ). Example 4. Consider the curve y = x 2 in the xy-plane. We can think of this as sitting inside R 3 as a curve if we look at { ( x,y,z ) | y = x 2 , z = 2 } . If we look at { ( x,y,z ) | y = x 2 } without specifying a z-value, then we get a surface (i.e. a thin film). We call the second shape a parabolic cylinder . 1.1. Distance. Definition. We write P ( x,y,z ) to mean a point P in R 3 with coordinates x,y,z . The distance between the points P 1 and P 2 is denoted | P 1 P 2 | . The formula for the distance in R 3 extends the formula for distance in the xy-plane: | P 1 P 2 | = p ( x 2- x 1 ) 2 + ( y 2- y 1 ) 2 + ( z 2- z 1 ) 2 Remember that the equation of a circle in the xy-plane could be interpreted as: (distance from a point) 2 = constant . The same is true for a sphere, i.e. the standard equation for the sphere of radius a centered at P ( x ,y ,z ) is ( x- x ) 2 + ( y- y ) 2 + ( z- z ) 2 = a 2 Example 5. Describe the region x 2 + y 2 + z 2- 2 x + 4 z ≤ 9 in R 3 . 1 2 MATH 214: INTRO TO VECTORS 2. Vectors First, we distinguish between vectors and points : Definition. A vector is a directed line segment between two points P 1 and P 2 , written-→ P 1 P 2 . P 1 is the initial point and P 2 is the terminal point . Two vectors are considered equal if they have the same length and direction. Note that the length of the vector-→ P 1 P 2 is the distance between P 1 and P 2 . By definition, if we replace P 1 by (0 , , 0) and P 2 by P 2 ( x 2- x 1 ,y 2- y 2 ,z 2- z 1 ), then v =-→ P 1 P 2 =-→ OP 2 . You can show that any vector is equivalent to one and only one vector with initial point O ....
View Full Document

## This note was uploaded on 07/31/2011 for the course MATH 214 taught by Professor Alexondrus during the Spring '11 term at University of Alberta.

### Page1 / 9

Ch12Notes - MATH 214: INTRO TO VECTORS 1. The Geometry of 3...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online