Math: Coordinate Geometry : Quantitative
Since this is a curve in the plane, it gives some relation between x and y; in fact, since it is a line that is Thus, all points on this graph have coordinates (x,0). Iterate over all coordinates (or a subset x-d,y-d x+d,y+d if the area is big). For each field of Use recursion along with a list/hash of visited links. Take a step. First we have plotted the data, with height in inches as the x coordinate and However, many points are off the line because the relationship is not all that close.
In the diagram above they are labeled Quadrant 1, 2 etc. It is conventional to label them with numerals but we talk about them as "first, second, third, and fourth quadrant".
Distance Between 2 Points
Point x,y The coordinates are written as an "ordered pair". The letter P is simply the name of the point and is used to distinguish it from others.
The two numbers in parentheses are the x and y coordinate of the point. The first number x specifies how far along the x horizontal axis the point is.
The Distance Formula
The second is the y coordinate and specifies how far up or down the y axis to go. It is called an ordered pair because the order of the two numbers matters - the first is always the x horizontal coordinate.
The sign of the coordinate is important. A positive number means to go to the right x or up y.
Negative numbers mean to go left x or down y. Distance between two points Given coordinates of two points, distance D between two points is given by: For a horizontal line, its length is the difference between the x-coordinates. For a vertical line its length is the difference between the y-coordinates. Distance between the point A x,y and the origin As the one point is origin with coordinate O 0,0 the formula can be simplified to: Find the distance between the point A 3,-1 and B -1,2 Solution: The midpoint of this line is exactly halfway between these endpoints and it's location can be found using the Midpoint Theorem, which states: In coordinate geometry, lines in a Cartesian plane can be described algebraically by linear equations and linear functions.
Every straight line in the plane can represented by a first degree equation with two variables. There are several approaches commonly used in coordinate geometry. It does not matter whether we are talking about a line, ray or line segment.
In all cases any of the below methods will provide enough information to define the line exactly. This form includes all other forms as special cases.
Points, Vectors and Coordinate Systems (why are points and vectors different?)
Using two points In figure below, a line is defined by the two points A and B. By providing the coordinates of the two points, we can draw a line.
No other line could pass through both these points and so the line they define is unique. Find the equation of a line passing through the points A 17,4 and B 9,9. Using one point and the slope Sometimes on the GMAT you will be given a point on the line and its slope and from this information you will need to find the equation or check if this line goes through another point. You can think of the slope as the direction of the line.
So once you know that a line goes through a certain point, and which direction it is pointing, you have defined one unique line. In figure below, we see a line passing through the point A at 14, With these two facts we can establish a unique line. This gives us a way to describe points. We need to have a starting point the origin.
Distance Between 2 Points
And then we can describe other points by the vectors that take you from this origin. So, a point is just a vector with a known origin.
Think about it this way: Coordinate Systems Having an origin is only part of what we need to interpret a point or vector.Ch. 2.2 Finding the Coordinates and Quadrants of Points Example 1
To interpret a vector, we basically need other vectors that will tell us how to interpret each of the numbers. These are actually vectors themselves. If you want to define a coordinate system something that tell us how to interpret a point in two dimensions, we need two basis vectors and an origin. So, if we agree that the origin is the south-west corner of the room, and our basis vectors are a step north and a step east, you can interpret 3,5 as a position: However, understanding these concepts is really important because it will help us generalize to other bases and coordinate systems.
A quick idea of where this is going… You could imagine that there are lots of possible bases. Above, I gave one for the screen, but I could have equally picked another one say, the origin is the center, and the first basis vector is from the center to the right edge, and the second basis vector is from the center to the top edge.
And this is the basis pardon the pun of linear algebra.