What is needed -by definition- to determine the shortest (Euclidean) distance between two parallel lines in the x,y plane? I am thinking if someone can write this down (mathematically) beautifully covering both knowing the equations of the lines or the coordinates of two points lying on the lines.
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Suppose there are two parallel lines: $w_1x_1+w_2x_2=c_1$ (Line 1) and $w_1x_1+w_2x_2=c_2$ (Line 2). What is the distance between them (the shortest distance between any two points)?  I know the answer is $d=\frac{|c_1-c_2|}{||w||}$ where $||w||=\sqrt{w_1^2+w_2^2}$.  The method I was going to calculate is as follows:  1) find any point on Line 1 $(x_1^0,x_2^0)$ such that $w_1x_1^0+w_2x_2^0=c_1$  2) calculate the perpendicular line (Line 3) to Line 1 and passing through $(x_1^0,x_2^0)$  3) find the point $(x_1^1,x_2^1)$ where Line 3 intersects Line 2  4) calculate the distance between $(x_1^0,x_2^0)$ and $(x_1^1,x_2^1)$  However I couldn't figure out the algebra of this method. Can someone show me the steps of the above calculation? Or is there any simpler way to calculate this? Thanks.
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I'm researching non-Euclidean geometry.  Now I am looking for a good source for it.  Please suggest me something.Thanks.
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The following show you the whole question.     Find the distance d bewteen two planes   \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray}   Find the other plane $C3\neq C1$ that has the distance d to the plane $C2$.   According to the example my teacher gave me, the answer should be :     Am I right? However, I do not know what is normal and why there are P(5) and Q($-\frac{1}{2}$).    Thank you for your attention
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I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric equation). Also find the distance between the beginning of axis and this plane.   I'm not really sure where to start. Any help would be appreciated.
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For a game I am developing I need an algorithm that can calculate intersections. I have solved the problem, but the way I have done it is really nasty and I am hoping someone here might have a more elegant solution.  A pair of points represent the end points of a line drawn between them. Given two pairs of points, do the drawn lines intersect, and if so, at what point?   So for example call the lines (A.x, A.y)-(B.x, B.y) and (C.x, C.y)-(D.x, D.y)  Can anyone think of a solution? A solution in any language will do.  Edit: A point I should have made clearer, the algorithm must return false if the point of intersection is beyond the lengths of the line segments.
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I have $(x_1, y_1), (x_2, y_2)$.  How do I find the point that's $d$ distance away from $(x_1, y_1)$ on a straight line to $(x_2, y_2)$?  I know I can get the length of the line with Pythagoras. I know if I drew a circle I could use the radius as distance and the point would be where the line and the circle intersect.  Could someone briefly explain each step to me please?  I don't understand
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I've drawn two random points on a two dimensional plane. If the distance between these two points is one unit, can a line of π units be drawn using a compass and straight edge?  Or, is there a proof that shows this is impossible?
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if we have two line segments in 3D, what would be the way to test whether these two lines are collinear or not? (I fogot to mentioned that my line segments are 3D. So, I edited the original post. Sorry for the inconveniences) I wish to check the direction of the lines and the perpendicular distance between them. Does these two factors are enough to decide whether 2 line segments are collinear or not. Thank you in advance.
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I need to create many parallel lines.   Is there a tool in ArcGIS Desktop where you input the distance between the parallels and the number of them?
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I have two arbitrary lines in 3D space, and I want to find the distance between them, as well as the two points on these lines that are closest to each other.  Naturally, this only concerns the skew case, since the parallel and intersecting cases are trivial.  I know how to find the distance, as the question was asked before and answered .  I haven't found a good explanation on how to find the two points that determine that distance, though.  So specifically, given two lines  $$L_1=P_1+t_1V_1$$ $$L_2=P_2+t_2V_2$$  I would like to find two points $X_1$ on $L_1$ and $X_2$ on $L_2$ such that the distance between $X_1$ and $X_2$ is minimal.
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Show that $n$ lines separate the plane into  $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.   I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0.  But I really have no idea how to begin the inductive step.  How do we know what k+1 we're supposed to arrive at?  Thanks!
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How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example:  Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm $$||f||=\sup_{t\in[0,1]}|f(t)|$$ from the function $f(t)=t^2$ to the linear hull of the functions $g(t)=t$  and  $h(t)=\sin(t)$  Any hints would be appreciated.
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I would like to find a point $d$ away on a line from a starting point only given a slope. My fist thought was to use the distance formula ($d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$), but I would have to solve for two variables. How can I solve find the point without graphing or brute forcing?
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Let $P(x, y, 1)$ and $Q(x, y, z)$ lie on the curves $$\frac{x^2}{9}+\frac{y^2}{4}=4$$ and $$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$ respectively. Then find the square of the minimum distance between $P$ and $Q$.  My Attempt is:  I tried to find minimum distance between the points $(-2,\sqrt{3})$ and $(6\cos \theta,4\sin \theta)$.
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I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book.  I understand the difference between two concepts when i'm working on non-euclidean space, but is there any even slight difference between these two concepts when it is $\mathbb{R}^k$?
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There is a set A which contains some discrete points (1-dimension), for example {1, 3, 37, 59}. And I want to pick one point from A which can minimize the sum of distances between this point and others.  For this "1-D line", many people say the median is the right point, but could anybody give me a concrete proof on it?
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Given a triangle in a plane with vertices A, B, C, find the point T that minimizes the sum of distances between A-T, B-T, and C-T.  I can experimentally determine this point by sampling the space and finding the minimum via an attractive, interactive web page:   Drag any vertex to see the gradient of value sums change and the point T be updated. I find it interesting to note that moving any vertex towards or away from T does not affect T, until the point where the vertex touches T and causes it to move along with it.  However, an experimental determination is neither efficient nor accurate, and prevents me from graphing other interesting properties such as the arcs of the vertices when a fixed sum is required.  Please help me find the equation for T (preferably in terms of vectors, as the answer should be the same in 2D or 3D).  Edit: Based on the answer, I've created an updated interactive using the calculated point and showing the construction: