Drawing perperdicular to existing coordinates

[VCVCVC]

Hi ,

I have drawn a line between two co-ordinates say (x1,y1) and (x2,y2). I have a point called (x3,y3) . From the point (x3,y3) i need to draw a perpendicular line to (x1,y1) and (x2,y2). so that i will be able to find the perpendicular distance . Is there any formula for calculating or drawing this perpendicular line. Any equations for finding this? I have depicted the question in terms of diagram which is attached along with this message.
pls help.


Thanks in advance

 

[HallsofIvy]

The slope of the line from (x₁,y₁) to (x₂,y₂) is m= (y₂-y₁)/(x₂-x₁). The slope of a line perpendicular to that is -1/m (the slopes of two perpendicular lines always have a product of -1). Finally, the line through (x₃,y₃) with slope -1/m is y= -1/m(x- x₃)+ y₃.

 

[VCVCVC]

Hi,

Thanks for the very prompt response. Will you please brief with one small example ? I am sorry and i am not much good in mathematics. your help will be greatly appreciated.

 

[Simon666]

The following is for three (or more) dimensions. If you have two points p1 and p2, the equation of a line going through p1 and p2 is generally given by:

p(t) = p1 + (p2-p1)*t

We normalize the direction vector (p2-p1) of this line and obtain the equation:

p(t) = p1 + e*t with e = (p2-p1)/||p2-p1||

Then the equation of the line perpendicular to |p1p2| going through p3 is given by:

p(t) = p3 + v3*t with v3 = (p3-p2) - (e*(p3-p2))*e

If you normalize v3 obtaining the normal vector e3 = v3/||v3||, the distance d3 of the point p3 to the line defined by p1,p2 is given by:

d3 = (p3-p2)*e3

With the dot product of two vector v1(x1,y1,z1) and v2(x2,y2,z2) given by:

v1*v2 = x1*x2 + y1*y2 + z1*z2

For two dimensions, just take the z component zero.

 

[HallsofIvy]

An example: Let (x₁,y₁)= (1, 1), (x₂,y₂)= (5,3) and (x₃,3₁)= (0,3). (Made up pretty much at random.)

The slope of the line through (1,1) and (5,3) is (3-1)/(5-1)= 2/4= 1/2. (In fact, it is easy to see that the line through those two points is given by y= (1/2)(x-1)+ 1.)

The slope of any line perpendicular to that is -2 (since -2(1/2)= 1). The line with slope -2 passing through (0,3) is y= -2(x-0)+ 3 or y= -2x+ 3.

To find the "perpendicular distance" you would now find the point where the two lines y= y= (1/2)(x-1)+ 1= (1/2)x+ 1/2 and y= -2x+ 3 intersect. That gives, of course, (1/2)x+ 1/2= -2x+ 3 or (2+ 1/2)x= 3- 1/2. That is, (5/2)x= 5/2 so x= 1 and then y= 1 (I swear! I picked the numbers "at random"!) Finally, the distance from (0,3) to (1,1) (the intersection) is sqrt(1²+ 2²)= sqrt(5).