Distance from a Point to a Line
The distance from a point to a line in mathematics is the shortest distance between the point and the line. It is calculated using the formula involving the coordinates of the point and the equation of the line. This concept is important in geometry and is used to determine the perpendicular distance from a point to a given line.
5 Key excerpts on "Distance from a Point to a Line"
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...You also need to know the special pairs of angles formed by parallel lines and a transversal and use the relationships to find missing angles. Distance Formula Given the two points (x 1, y 1) and (x 2, y 2), the distance between these two points is given by the formula: The subscripts indicate that there is a first and second point. It does not matter which point is first and which one is second, the answer will be the same. Example: Find the distance between the points (0, −3) and (−2, 4). Using (0, −3) as (x 1, y 1) and (−2, 4) as (x 2, y 2), substitute into the formula. The distance between the points (0, −3) and (−2, 4) is approximately 7.28. Example: Find the length of. Identify the coordinates of point A and point B. Using (−3, −1) as (x 1, y 1) and (3, 2) as (x 2, y 2), substitute into the formula. The distance between the points (−3, −1) and (3, 2) is approximately 6.71. Midpoint Formula Given the two points (x 1, y 1) and (x 2, y 2), the midpoint between these two points is given by the formula: The subscripts indicate that there is a first and second point. It does not matter which point is first and which one is second, the answer will be the same. Example: Find the midpoint of a segment with endpoints (−4, −6) and (0, −8). Using (−4, −6) as (x 1, y 1) and (0, −8) as (x 2, y 2), substitute into the formula. The midpoint of a segment with endpoints (−4, −6) and (0, −8) is (−2, −7). Example: Find the midpoint of. Identify the coordinates of point E and point F. Using (−2, 2) as (x 1, y 1) and (1, −2) as (x 2, y 2), substitute into the formula. The midpoint of a segment with endpoints (−2, 2) and (1, −2) is. Example: On a coordinate plane, has endpoint Y at (8, 4). The midpoint of is M (2, −1)...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...4 Analytic Geometry 1. Rectangular Coordinates The points in a plane may be placed in one-to-one correspondence with pairs of real numbers. A common method is to use perpendicular lines that are horizontal and vertical and intersect at a point called the origin. These two lines constitute the coordinate axes; the horizontal line is the x-axis and the vertical line is the y-axis. The positive direction of the x-axis is to the right whereas the positive direction of the y-axis is up. If P is a point in the plane one may draw lines through it that are perpendicular to the x- and y-axes (such as the broken lines of Figure 4.1). The lines intersect the x-axis at a point with coordinate x 1 and the y-axis at a point with coordinate y 1. We call x 1 the x-coordinate or abscissa and y 1 is termed the y-coordinate or ordinate of the point P. Thus, point P is associated with the pair of real numbers (x 1, y 1) and is denoted P (x 1, y 1). The coordinate axes divide the plane into quadrants I, II, III, and IV. FIGURE 4.1. Rectangular coordinates. 2. Distance between Two Points; Slope The distance d between the two points P 1 (x 1, y 1) and P 2 (x 2, y 2) is d = (x 2 − x 1) 2 + (y 2 − y 1) 2 In the special case when P 1 and P 2 are both on one of the coordinate axes, for instance, the x-axis, d = (x 2 − x 1) 2 = | x 2 − x 1 |, or on the. y-axis, d = (y 2 − y 1) 2 = | y 2 − y 1 |. The midpoint of the line segment P 1 P 2 is (x 1 + x 2 2, y 1 + y 2 2). The slope of the line segment P 1 P 2, provided it is not vertical, is denoted by m and is given by m = y 2 − y 1 x 2 − x 1. The slope is related to the angle of inclination α (Figure 4.2) by m = tan α Two lines (or line segments) with...
eBook - ePubFigures of Thought
Mathematics and Mathematical Texts
- David Reed(Author)
- 2003(Publication Date)
- Routledge(Publisher)
...As points define the delimiting of lines, straight lines are precisely those for which no additional specification of the relationship between points-as-extremities and lines-as-delimited is either necessary or possible. For straight lines this relationship is always the same. No metric or measure theoretic specification is implied here. The delimitation happens in the same manner throughout the line, but nothing is said about what this manner may be. 8 Definitions 2, 3 and 4 taken together provide the paradigm in Euclid's mathematics of a ‘measured thing’: • that which is measured (in this case lines) is defined in terms of a measurable (in this case length); • the measurable is further determined by specifying how the measuring or delimitation is to be performed (in this case by points-as-extremities) and this specification requires nothing further (there is nothing further to say about points); • a special type or kind of measured thing (in this case straight lines) is defined by requiring that the delimitation of the measurable does not vary with what is delimited and that no further specification of this relationship is necessary. This must be understood as a formal procedure which moves mechanically step-by-step to define and determine the measured thing. Once the starting point (!) is adopted there is no opportunity for variation or deviation. This is Euclid's approach to ‘rigour’ in mathematics and failure to understand his method leads to the view that some ‘selection’ process based on tradition or philosophical inclination led Euclid to pick the particular formulations and sequence of definitions that we find in the Elements. 9 The importance of a clear understanding of the nature of Euclid's argument goes beyond an appreciation of the clarity and cogency of his thinking; it is fundamental to understanding his subject matter...
eBook - ePub- Stephen Wise(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...In the sieve of Eratosthenes example, we just needed a list of whether each number from 1 to 100 was prime or not. To store the data for points, we need to store three pieces of information for each point – the ID, the X coordinate and the Y coordinate. This is not a problem, however – we simply add extra dimensions to the array: POINTS[1..100,1..3] This declares an array which has 100 rows each of which has three columns. To store the data for the first point, we could use POINTS[1,1] = 1 POINTS[1,2] = 4.5 POINTS[1,3] = 2.5 It then becomes possible to use these coordinates to do calculations such as finding the distance between two of the points. To do this, we use one of the best-known expressions in mathematics, the Pythagoras’ equation. This states that for any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. With the sides labelled a, b and c as in Figure 3.3, this can be expressed as a 2 + b 2 = c 2 (3.1) This equation can be rearranged to give c in terms of a and b c = a 2 + b 2 (3.2) FIGURE 3.3 Illustration of how distance can be calculated using Pythagoras’ theorem. If you look at Figure 3.3, you can see that the length of the side labelled a can be calculated by taking x1 from x 2. In the same way, b is the difference between y 1 and y 2. So, if we want to know the distance between these two points, the equation is d = (x 1 − x 2) 2 + (y 1 − y 2) 2 (3.3) If we wish to calculate the distance between point 1 and 25 then in pseudocode this becomes Distance = sqrt(power(POINTS[1,2] - POINTS[25,2],2) +power(POINTS[1,3] – POINTS[2 5,3],2)) We can use the same basic equation to work out the length of a line by simply working out the distance between each pair of points and adding these up...
eBook - ePub- Thomas D. Wickens(Author)
- 2014(Publication Date)
- Psychology Press(Publisher)
...Chapter 2 Some vector geometry A geometric vector gives a concrete representation to a variable and to the algebraic vector of data that measures it. Before applying this geometric representation to the techniques of multivariate statistics, one needs to understand how to manipulate these vectors and combine them. The first section of this chapter describes these operations, and the second section considers the correspondence between vectors and variables. The final two sections describe the important concepts of vector spaces, linear dependence, and projection. 2.1 Elementary operations on vectors Operations such as addition and multiplication that apply to algebraic vectors have their counterparts for geometric vectors. Vectors. A vector is a directed line segment. It has two properties, its direction and its length. It can be started from any point. Both the vector that goes from the point (0, 0) to the point (2, 1) and the vector that goes from (3, 2) to (5, 3) move two units over and unit one up, and so are different instances of the same vector: The standard position from which to start a vector is at the origin 0. However, when combining several vectors or illustrating the relationships among them, it can be helpful to start them at different points. As long as neither its direction nor length is altered, one can freely slide a vector about a diagram without changing it. The length of a vector. A fundamental property of any vector is its length, which is denoted by placing vertical bars about it—the length of the vector is | |. Algebraically, the length of a vector is found by using the Pythagorean theorem. The vector = [ x 1, x 2 ]′ has length. More generally, the length of the n -dimensional vector = [ x 1, x 2, …, x n ]′ is For example, the length of the vector = [2, 1]′ in the diagram above is Scalar multiplication. The simplest operation that changes one vector into another is multiplication by a number such as 2 or − 7...
