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How to find intersection of lines? Snehal Poojary.

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1 How to find intersection of lines? Snehal Poojary

2 Lines that have one and only one point in common are known as intersecting lines. If they are in the same plane there are three possibilities  if they have more than one point in common which means they are the same line  if they are distinct but have the same slope they are said to be parallel and have no points in common  otherwise they have a single point of intersection If two lines are not in the same plane they are called skew lines and have no point of intersection

3 When straight lines intersect on a 2-dimensional graph, they meet at only 1 point, which can be described by a single set of x- and y-coordinates Since both lines pass through that point, you know that the x- and y- coordinates must satisfy both equations To find out where they intersect you need to solve a system of two equations with two variables

4 Line 1 ax + by = c (a,b). (x,y) = c Line 2 dx + ey = f(d,e). (x,y) = f Each line equation can be expressed as Bw=m [a,b] [x] = [c] [d,e] [y] = [f] So this is a linear system involving a 2x2 matrix, the matrix “B” = [a,b] [d,e] The right hand side vector “m” = [c] [f] The point of intersection is “w” = [x] [y]

5 y = x + 3 (line 1) y = 12 – 2x (line 2) x + 3 = 12 – 2x x + 2x = 12 – 3 3x = 9 x = 3 Replace the value of x with 3 to solve y y = x + 3 (line 1) and now x = 3 y = 3 + 3 y = 6 So now we have the coordinate for where the 2 lines intersect (3, 6)

6 n-line intersection In 2 dimensions to find an intersection between n number of lines, write the i-th equation (i = 1,...,n) as (b i1 b i2 )(x y) T = m i, and stack these equations into matrix form as Bw=m If B has independent columns, its rank is 2. Then if and only if the rank of the augmented matrix [B|m] is also 2, there exists a solution of the matrix equation and thus an intersection point of the n lines. The intersection point, if it exists, is given by B T Bw=B T m w = (B T B) -1 B T m

7 In 3 dimensions a line is represented by the intersection of two planes, each of which has an equation of the form (b i1 b i2 b i3 )(x y z) T = m i, Thus a set of n lines can be represented by 2n equations in the 3-dimensional coordinate vector w = (x, y, z) T Bw=m Where now B is 2n × 3 and m is 2n × 1. As before there is a unique intersection point if and only if B has full column rank and the augmented matrix [B | m ] does not, and the unique intersection if it exists is given by w = (B T B) -1 B T m

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