Given a line
#L -> p = p_1 + lambda vec v# and a plane
#Pi-> << p-p_1, vec n >> = 0#
if #L in Pi rArr lambda << vec v, vec n >> = 0#
now given a sphere
#Sigma-> norm(p-p_0) = r#
the normal vector to #Sigma# is
#vec sigma = (p-p_0)/norm(p-p_0)#
then a tangent plane to #Sigma# should obey
#{(<< p-p_1, (p-p_0)/norm(p-p_0) >> = 0),(<< vec v , (p-p_0)/norm(p-p_0) >> = 0),
(norm(p-p_0) = r):}#
or
#{(normp^2 - << p, p_0+p_1 >> + << p_0, p_1 >> = 0),
(<< vec v, p >> = << vec v, p_0 >> = 0),
( norm(p-p_0) = r):}#
here
#p_0 = (-1,2,-3)#
#r = sqrt 21#
#p_1 = (13/3, 1,-2/3)#
#vec v = (3,6,0)#
and after solving we get
#t_1 = (1,1,1)# and
#t_2 = (3,0,-4)#
the two tangent points, so the planes are
#Pi_1-> << p-p_1, vec sigma_1 >> = 0#
#Pi_2-> << p-p_1, vec sigma_2 >> = 0#
with
#vec sigma_1 = (2,-1,4)#
#vec sigma_2=(4,-2,-1)#
