Mathematics
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.
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4 Key excerpts on "Distance from a Point to a Line"
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
As an exercise, carefully draw the graphs of the two lines in the previous example. Try to determine the coordinates of the point where the lines intersect.2.3.5The Distance between Points on a Line
If a line is horizontal, it has the equation y = c, where c is the y-intercept. The distance between points P1 (x1 , c) and P2 (x2 , c) (denoted |P1 P2 |) is the difference in the x-coordinates, that is, |P1 P2 |=|x2 − x1 |. (We write this as an absolute value because, in general, x2 need not be larger than x1 .) If a line is vertical, it has the equation x = c, where c is the x-intercept of the line. The distance between points Q1 (c, y1 ) and Q2 (c, y2 ) is the difference in the y-coordinate, that is, |Q1 Q2 | = |y2 − Y1 | (see figure 2.9 ).FIGURE 2.9. Horizontal and vertical lines.If a line is neither vertical nor horizontal, it has the equation y = mx + c, where m ≠ 0. In this case, the distance between points P1 (x1 , y1 ) and P2 (x2 , y2 ) is determined by means of the Pythagorean theorem (theorem 9.5.8(a) ) applied to the right triangle shown in figure 2.10 . The length of the horizontal side of the right triangle is |P1 P| = |x2 − x1 |, and the length of the vertical side of the right triangle is |P2 P | = | y2 − y1 |. Thus, we have:FIGURE 2.10. The distance between two points on a line.Formula (2.7) is used to find the distance between any two points in the Cartesian plane.EXAMPLE 2.3.11. The distance between the points A(−3, 6) and B(5, 1) is2.3.6The Equation of a Perpendicular Bisector
We find the midpoint of the line segment joining two points P1 (x1 , y1 ) and P2 (x2 , y2 ) by computing the average of the x- and y-coordinates to produce the pointDEFINITION 2.3.5. The perpendicular bisector of a line segment is the line that passes through the midpoint of the segment and is perpendicular to it.EXAMPLE 2.3.12. Given points A(−3, 1) and B(5, 4), find the equation of the perpendicular bisector l of the line segment AB
eBook - ePub- Steven J. Janke(Author)
- 2014(Publication Date)
- Wiley(Publisher)
Projecting a vector (from the line to the plane) onto the normal vector again gives the distance between the line and the plane. Example 3.8 (Distance between Two Parallel Planes) We have two parallel planes with a common normal vector. One plane contains the point and the other contains the point. To find the distance between the planes, we calculate the coordinates of and then project it onto the normal vector. Notice that the order we subtract the points to get does not matter (Figure 3.8). Figure 3.8 Distance between parallel planes If we are thinking about computation efficiency, notice that, if we normalize the vector so that it has length, then no division is required to calculate the projection and hence the distance. Of course, it takes three divisions to normalize a vector, but if we will be using it many times, it might be worth it. Example 3.9 (Distance from a Line to a Plane) Instead of two planes, suppose we have one containing the point with normal. Consider now the line. The direction vector is perpendicular to the normal and hence parallel to the plane. By projecting a vector from to a point on the line, say, we can find the distance between the line and the plane. This time, normalize the vector first, giving In this case, picking any point on the line gives the same distance. If the line was not parallel to the plane, this would not be true. 3.2.4 Line to a Line In two dimensions, two lines either intersect or they are parallel. In three dimensions, it is also possible that they do not intersect and they are not parallel; these are skew lines. Suppose that for two nonintersecting lines we have found exactly where the lines come closest to each other. That is, we have a point on the first line and a point on the second line such that the distance between the two points is as close as possible for any points on the lines. Then the vector must be perpendicular to each line. If it is not, then suppose it is not perpendicular to the second line
eBook - ePubNew-Technology Flowmeters
Volume I
- Jesse Yoder(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
Since flow seems closely connected to continuity, it is worth looking at the idea of continuity. The number line itself is continuous, and yet many mathematicians view the number line as being made up of discrete points. What is confusing about this analysis is that points have no area. This idea follows the assumption that it is always possible to fit another point between any two points. However, if points have no area, meaning they do not have width but are essentially dimensionless, then 1,000 points or one million points will also not have area. Mathematicians compensate for this by using the idea of infinity, arguing that even if points have no area, surely infinitely many of them will have area. But infinity multiplied by 0 is still 0 and adding infinity to dimensionless points does not yield width, length, or the number line, which is continuous.Part of the problem with this reasoning is that a line is made up of points. But if points lie on the line instead of being a part of the line, then the line can have continuity independent of the points. In fact, one way of conceiving of a line is as the path of a point in motion. Likewise, a plane is formed by placing a line in motion.A Line Is the Path of a Moving Point
What is the relationship between points and a line? A line is the path of a moving point, as Aristotle says in De Anima 1:4. Likewise, a plane is the path of a moving line. A point and a line then are intimately related, but not in the way Euclidean geometry describes them as being related. A line is somewhat like the trail of a meteor, except that when we use the point of a pencil to draw a line, the line is static and remains visible.Points Lie on the Line, Not in the Line
Anyone who is aware of our language will realize that we speak of points being on a line more naturally than of points being in a line. The idea that points are in a line is more a result of mathematical analysis than of an understanding of mathematical language. But what is the difference between points being on a line and points being in a line?Someone who is sitting on a fence is not part of the fence; instead, his or her body is physically touching the fence. But no one would think that a person sitting on a fence is part of the fence. Instead, the fence is made up of steel, wood, rocks, or some other material, depending on what type of fence it is. Likewise, a book lying on a table is not part of the table, although the book touches the table.
eBook - ePub- Fletcher Dunn, Ian Parberry(Authors)
- 2011(Publication Date)
- A K Peters/CRC Press(Publisher)
The zero vector cannot be normalized. Mathematically, this is not allowed because it would result in division by zero. Geometrically, it makes sense because the zero vector does not define a direction—if we normalized the zero vector, in what direction should the resulting vector point?2.9.2 Geometric Interpretation
In 2D, if we draw a unit vector with the tail at the origin, the head of the vector will touch a unit circle centered at the origin. (A unit circle has a radius of 1.) In 3D, unit vectors touch the surface of a unit sphere. Figure 2.16 shows several 2D vectors of arbitrary length in gray, beneath their normalized counterparts in black.Notice that normalizing a vector makes some vectors shorter (if their length was greater than 1) and some vectors longer (if their length was less than 1).Figure 2.16 Normalizing vectors in 2DWe are now prepared to derive one of the oldest and most fundamental formulas in computational geometry: the distance formula. This formula is used to compute the distance between two points.2.10 The Distance Formula
First, let’s define distance as the length of the line segment between the two points. Since a vector is a directed line segment, geometrically it makes sense that the distance between the two points would be equal to the length of a vector from one point to the other. Let’s derive the distance formula in 3D. First, we will compute the vector d from a to b . We learned how to do this in 2D in Section 2.7.3 . In 3D, we used = b - a =[.]b x−a xb y−a yb y−a zThe distance between a and b is equal to the length of the vector d , which we computed in Section 2.8 :distance ( a , b ) =‖ d ‖=.+d x2+d y2d z2Substituting for d , we getdistance ( a , b ) =‖ b - a ‖=(.b x-a x) 2+ (b y-a y) 2+ (b z-a z) 2The 3D distance formulaThus, we have derived the distance formula in 3D. The 2D equation is even simpler:
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