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How to find intersection of lines? Snehal Poojary

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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

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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

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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]

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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)

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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

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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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