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)