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Distance from a point to a line - 3-Dimensional.

Distance from a point to a line

Definition.
Distance from a point to a line — is equal to length of the perpendicular distance from the point to the line.

Distance from a point to a line in space formula

If M0(x0, y0, z0) point coordinates, s = {m; n; p} - directing vector of line l, M1(x1, y1, z1) - coordinates of point on line l, then distance between point M0(x0, y0, z0) and line l can be found using the following formula:

d |M0M1×s|
|s|

Proof of the formula of distance from a point to a line for the space problem

If l is line equation then s = {m; n; p} - directing vector of line, M1(x1, y1, z1) - coordinates of point on line. From properties of cross product it is known that the module of cross product of vectors is equal to the area of a parallelogramme constructed on these vectors

S = |M0M1×s|.

On the other hand parallelogramme area is equal to product of its side on height spent to this side

S = |s|d.

Having equated the areas it is simple to receive the formula of distance from a point to a line.


Examples of tasks with from a point to a line in space

Example 1.
To find distance between point M(0, 2, 3) and line

x - 3  =  y - 1  =  z + 1
2 1 2

Solution.

From line equation find:

s = {2; 1; 2} - directing vector of line;
M1(3; 1; -1) - coordinates of point on line.

Then

M0M1 = {3 - 0; 1 - 2; -1 - 3} = {3; -1; -4}

M0M1×s i j k  = 
  3    -1    -4  
  2    1    2  

= i ((-1)·2 - (-4)·1) - j (3·2 - (-4)·2) + k (3·1 -(-1)·2) = {2; -14; 5}

d |M0M1×s|  =  22 + (-14)2 + 52  =  225  =  15  = 5
|s| 22 + 12 + 22 9 3

Answer: distance from point to line is equal to 5.



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