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Distances
Point-Line Distance

The distance from a point, P, to a line, r, is the smallest distance from the point to one of the infinite points on the line. The distance corresponds to the perpendicular line from the point to the line.

 The distance from a point, P, to a line,r, is defined by:

Determine the distance from the point P=(1,3,-2) to the line

 x=2+3λ y=-1+λ z=1-2λ

Solution:

$\vec{AP}$=(1-2,3+1,-2-1)=(-1,4,-3)

$\vec{u_r}$=(3,1,-2)

$\vec{u_r}$ x $\vec{AP}$ =
 $\vec{i}$ $\vec{j}$ $\vec{k}$ 3 1 -2 -1 4 -3
= 5$\vec{i}$+11$\vec{j}$+13$\vec{k}$

$|\vec{u_r}\;x\;\vec{AP}|\;=\sqrt{5^2+11^2+13^2}=\;3\sqrt{35}$

$|\vec{u_r}|=\;\sqrt{3^2+1^2+(-2)^2}=\;\sqrt{14}$

$d(P,r)=\;\frac{3sqrt{35}}{sqrt{14}}\;=\;\frac{3sqrt{10}}{2}\;$

Determine the distance from the point P=(1,2,3) to the line

$r\;\equiv\;\frac{x-2}{4}=\frac{y-3}{4}=\frac{z-4}{2}\;$

Solution:

A=(2,3,4)

$\vec{AP}$=(1-2,2-3,3-4)=(-1,-1,-1)

$\vec{u_r}$=(4,4,2)

$\vec{u_r}$ x $\vec{AP}$ =
 $\vec{i}$ $\vec{j}$ $\vec{k}$ -1 -1 -1 2 2 1
= -2$\vec{i}$+2$\vec{j}$

$|\vec{u_r}\;x\;\vec{AP}|\;=\sqrt{(-2)^2+2^2+0^2}=\;\sqrt{8}$

$|\vec{u_r}|=\;\sqrt{4^2+4^2+2^2}=\;6$

$d(P,r)=\;\frac{2sqrt{2}}{6}\;=\;\frac{sqrt{2}}{3}\;$