Define the unit vector ˆeα=|α^1 |αand construct the line from (x 0 , y 0 , z 0 )which is
perpendicular to the given line and label this distance d. Our problem is to find the
distance d. From the geometry of the right triangle with sides r 0 −r 1 and done can
write sin θ=|r d
0 −r 1 |. Use the fact that by definition of a cross product one can write
|(r 0 −r 1 )׈eα|=|r 0 −r 1 ||ˆeα|sin θ=d=∣∣
∣∣(r 0 −r 1 )× α
|α|∣∣
∣∣Example 6-14. Find the equation of the plane which passes through the point
P 1 (x 1 , y 1 , z 1 )and is perpendicular to the given vector N =N 1 ˆe 1 +N 2 eˆ 2 +N 3 ˆe 3.
Solution Let r 1 =x 1 ˆe 1 +y 1 ˆe 2 +z 1 ˆe 3 denote the
position vector to the point P 1 and let the vec-
tor r =xˆe 1 +yˆe 2 +zeˆ 3 denote the position vector
to any variable point (x, y, z )in the plane. If the
vector r −r 1 lies in the plane, then it must be
perpendicular to the given vector N and conse-
quently the dot product of (r −r 1 )with N must
be zero and so one can write
(r −r 1 )·N = 0 (6 .43)as the equation representing the plane. In scalar form, the equation of the plane is
given as
(x−x 1 )N 1 + (y−y 1 )N 2 + (z−z 1 )N 3 = 0 (6 .44)Example 6-15. Find the perpendicular distance dfrom a given plane
(x−x 1 )N 1 + (y−y 1 )N 2 + (z−z 1 )N 3 = 0to a given point (x 0 , y 0 , z 0 ).