The equation of a line in slope-intercept form is given by
(1)
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so the line has slope . Now consider the distance from a point to the line. Points on the line have the vector coordinates
(2)
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Therefore, the vector
(3)
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is parallel to the line, and the vector
(4)
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is perpendicular to it. Now, a vector from the point to the line is given by
(5)
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Projecting onto ,
(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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If the line is specified by two points and , then a vector perpendicular to the line is given by
(12)
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Let be a vector from the point to the first point on the line
(13)
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then the distance from to the line is again given by projecting onto , giving
(14)
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As it must, this formula corresponds to the distance in the three-dimensional case
(15)
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with all vectors having zero -components, and can be written in the slightly more concise form
(16)
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where denotes a determinant.
The distance between a point with exact trilinear coordinates and a line is
(17)
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(Kimberling 1998, p. 31).