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

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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 ....
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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.

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

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