Kabissa

Transcription

Three Dimensional Coordinate Systems (Level 1) Calculus III is a course that uses all the concepts learned from calculus I and extends it to the study of functions with two or more variables. In a nutshell calculus III is calculus I in 3 dimensions! Nothing more nothing less. With that said this class will challenge your spatial visualization skills and push them to the limit. Before we dive in we need to start thinking 3 dimensionally and be familiar with the 3-dimensional coordinate System. We first need to remember the basics of number lines. Back in Algebra I and perhaps even before that, you learned that a number line is formed when a numerical value is assigned to each point on a line. For example the following line represents a number line and point A is located at -1, likewise point B is located at 1 and half or written another way as 1.5. Notice that number lines are useful to describe the location of a point relative to the origin which is usually assigned a value of zero. In other words, a number line can be viewed as a coordinate system, specifically a one dimensional coordinate system consisting of all the points on a straight line. Notice that this line is broken into two regions or half's the first half contains positive numbers and the second half contains negative numbers and for the most part you can only move along the line either infinitely to the left or infinitely to the right. In this sense, a 1-dimensional coordinates system is equivalent to a number line where every real number, whether rational or irrational, has a unique location or position on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum that includes the real numbers. We can mathematically denote this 1-dimensioanl space that includes the set of all real numbers as follows . Using the real number symbol. Now if we take this number line and introduce an additional number line that is perpendicular to this number line we obtain the modern coordinate system, also known as the Cartesian coordinate system or rectangular coordinate system. A two dimensional coordinate system specifies each point uniquely in a plane also known as the xy plane, as oppose to a 1 dimensional coordinate system which specifies each point in a straight line. As a result, a point on a two dimensional coordinate system is described by a pair of numerical coordinates also known as an ordered pair, each coordinate in the order pair describes the location of the point along each number line which are commonly known as coordinate axis, The value along the x-axis also known as the abscissa is called the x-coordinate of the ordered pair and is denoted first when describing a point on the plane. Likewise the value, along the y axis also known as the ordinate is called the y-coordinate and is denoted second when describing a point together the x and y coordinate form an ordered pair. Notice that a point is measured relative to a common center where the two number lines intersect this common point of intersection is called the origin and is usually labeled by using the letter O the ordered pair that describes this point is written as follows (0,0), as an example the following 4 points can be described as follows: point A is located at (2,2) point B is located at (-1,1) point C is located at (-2,-3) and point D is located at (4,-3). As you can see each point forms a rectangle when plotting their position hence the rectangular coordinate system. Also Notice that a two dimensional coordinate system is broken up into 4 regions also known as quadrants, they are usually labeled as follows quadrant I quadrant II quadrant III and quadrant IV, in addition the sign for each coordinate of an ordered pair is dependent on which quadrant the point is located. If the point is located in quadrant I both the x and y coordinate will be positive, in quadrant II the x coordinate will be negative and the y coordinate will be positive, in quadrant III both the x and the y coordinate will be negative, and finally in quadrant IV the x-coordinate will be positive and the y-coordinate will be negative. In a two dimensional coordinate system you are free to move along two lines, infinitely to the left, infinitely to the right, infinitely up and infinitely down. As oppose to just one line like that in a one dimensional coordinate system. Mathematically we can define all the points in a two dimensional coordinate system by the Cartesian product R times R or R squared where R squared represents the xy-plane and it consist of all the ordered pairs where the values of x and y are elements of the real numbers. Let's move along and talk about the Three Dimensional Coordinate System also known as a three dimensional rectangular coordinate system. If we take the two dimensional Cartesian axis and add an additional number line Perpendicular to both axis we obtain a three dimensional coordinate system. The three dimensional coordinate system contains 3 coordinate axes that intersect at a common point also called the origin, these include the x-axis, y-axis and z-axis. For the most part the x and y axis are horizontal lines and the z axis is a vertical line. Just like the previous coordinate systems, each axis contains positive values and negative values, on the x axis the positive values are located here, and the negative values are located on the opposite direction. On the y axis the positive values are located here, and the negative values in the opposite direction, and finally on the Z axis the positive values are located here and negative values in the opposite direction. A way for you to remember where the direction of the z axis is located is by using the right hand rule, if you curl the fingers of your right hand around the z-axis counter clock wise from the positive x-axis to the positive y-axis, then your thumb points towards the positive direction of the z-axis. In addition the three coordinate axes also determine 3 coordinate planes. The xy plane is the plane that contains the x and y axes; the yz plane contains the y and z axes, and the xz plane contains the x and z axes. Together the three coordinate planes divide space into eight regions called octants. This is analogous to the quadrants of a two dimensional coordinate system, in this case a three dimensional coordinate system divides space into octants. The first octant is determined by the positive axes, if you are having a hard time visualizing this just look at any bottom corner of a room, that's essentially a visual representation of the first octant. The wall on your right is the yz plane the wall on your left is the xz plane and the floor is the xy plane. In the same manner there are 7 other rooms 3 more on top and 4 more on the bottom. If you imagine this three dimensional coordinate system representing a two story building there would be 4 rooms on top and 4 rooms on the bottom. Each room sharing a common corner called the origin. In a three dimensional coordinate system a point is located in space as oppose to a two dimensional system where a point is located on a plane and a 1 dimensional system where a point is located on a line. With that said, we represent a point by an ordered triple x, y, z of real numbers and we call the numbers the x, y and z coordinates of the point. In order to locate a point in space we first start at the origin and move along the x axis to locate the first coordinate then move parallel to the y axis for the second coordinate and then parallel to the z axis to locate the last coordinate. The reason why this coordinate system is called the rectangular coordinate system is because points in space determine rectangular boxes. For example the coordinate (4,5,3) is plotted as follows, we first move along the x axis 4 units in the positive direction then we move 5 units parallel to the y axis in the positive direction, and then move parallel along the z axis also in the positive direction. Now at first it's a little hard to see where the point is located so we need to draw a corresponding box so we can get a better perspective and visualize this 3 dimensional plot on a 2 dimensional sheet of paper. The way we draw the box is actually pretty easy all we need to do is draw 3 additional lines with the same length for each coordinate. So here we draw 2 lines parallel to the first coordinate that starts from the point (0,5,0) and ends at (4,5,0) another one that starts from (0,5,3) and ends at (4,5,3) and lastly one that starts from (0, 0, 3) amd ends at (4,0,3). Then we draw lines parallel to the y coordinate one for the bottom and two for the top of the box, then we do the same for the z coordinate we draw lines parallel to connect the bottom and the top of the box. In the same way, the following points would be plotted as follows: point B would be located at (4,-6,6) point C would be located at: (-3,-3,2) point D would be located at: (-6,3,4) point E would be located at: (4,5,-5) point F would be located at: (4,-6,-5) point G would be located at: (-3,-3,-2) and point H would be located at: (-6,3,-4) This is how the points would look like in a sheet of paper, lets actually plot these points using a 3D plotter Point A would look like this Point B Point C Point D Point E Point F Point G and finally point H so we graph this Cartesian plane and move it around like this we see that we have 8 distinct points four off them on top of the xy plane and four of them on the bottom of the xy plane one for each octant it is really hard to see this by just using a two dimensional sheet of paper but by using technology we can actually see a better representation of how these points look like in three dimensional space. Just like the quadrants of a two dimensional coordinate system determines the sign of each coordinate so does each octant, on the first Octant all 3 coordinates are positive, in the second octant the x axis negative and both the y and z axis are positive, in the third octant both x and y are negative and z is positive, in the 4th octant both x and z are + and y is -, in the 5th octant both x and y are + and z is -, in the 6th octant x and z are - and y is +, in the 7th octant, all 3 are negative, and finally in the 8th octant x is + and both y and z are -. In a three dimensional coordinate system you are free to move along three lines, infinitely to the west, infinitely to the east, infinitely north, infinitely south, infinitely up and infinitely down. Mathematically we can define all the points in a three dimensional coordinate system by the Cartesian product R times R times R or R cubed where R cubed in this case 3d space consist of all the ordered triples where the values of x y and z are elements of the real numbers. Alright make sure you are comfortable with this new 3 dimensional coordinate system it's going to be key in understanding the rest of calculus III, in our next video we will focus on graphing equations using this new coordinate system.