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by confused on March 6th, 2008

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What is the distance between the parallel lines of y=2x + 4 and y=2x + 1? How do you do this?

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Answers. 3 helpful answers below.

  • by jehan60188 on March 6th, 2008

    jehan60188

    "High school oriented solution"

    a point on the first line: (0,4)
    draw a line *perpendicular* to both lines
    y = -(1/2)x + 4

    (0,4) is also on that line
    that line also intersects the second line:
    2x+1 = -(1/2)x + 4
    2x = -(1/2)x + 3
    2.5 x = 3
    x= 1.2

    and y = -(1/2)*1.2+4 = 3.4

    so, starting from (0,4) and going perpendicularly towards line two, we would hit (1.2, 3.4); why is it important to go perpendicular?
    so now, we need the distance between (0,4) and (1.2, 3.4)

    distance formula

    sqrt(1.2 ^2 + (-.6)^2) = 1.34164079
    and THAT is the distance between the two lines!

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  • by jehan60188 on March 6th, 2008

    jehan60188

    from: http://www.answerbag.com/q_view/606046

    vector calc based solution:

    first we define the lines
    L1: p1 + t*v1
    L2: p2 + u*v1

    where pi is a point, and v1 is a vector and t & u are time
    notice I only use v1, NOT v1 & v2. That's because it's the same vector (parallel lines)

    not this problem turns into finding the distance between p1 and L2
    the smallest distance would be the length of a perpendicular from p1 to L2, intersecting at a point, g

    draw a (right) triangle with (p2,p1) being the hypotenuse, and (p1,g), (p2,g) beings legs
    we know (by pythagorous)
    (p2,p1)#(p2,p1) = (p1,g)#(p1,g) + (p2,g)#(p2,g), where '#' is being used as the dot operator
    (p2,p1) is known, since we defined the lines with that
    also, (p2,g) is the 'projection' of (p2,p1) onto v1- so to find it's distance take (p2,p1)#v@
    where v@ is the 'unit vector' of v1- or more simply: v@ = v1/||v1||

    now you have enough to solve for (p1,g)#(p1,g)!

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  • by Mr. Meaulnes on March 6th, 2008

    Mr. Meaulnes

    As far as I can tell, the best way would be to graph it, find two points (one on each line) that line up vertically on the y axis, and then count out the horizontal distance between them on the x axis.

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