Equation of plane
a(x - x1) + b(y - y1) + c(z - z1) = 0
given line
r = -3i + j + 5k + 3A i - Aj -5Ak
taking A = 1
r = 0, thus (0, 0, 0) is a point on the line
Take A = 2
r = 3i - j - 5k
(3, -1, -5) is also a point on the line
hence the plane passes through three points (0,0,0) , (1, 1, 1) and (3, -1,-5)
a(1 - 0) + b(1 - 0) + c(1 - 0) = 0
a + b + c = 0
a(3 - 0) + b(-1 - 0) + c(-5 - 0) = 0
3a - b - 5c = 0
a / (-5 + 1) = -b/(-5 - 3) = c/(-1 - 3)
a/-4 = b/8 = c/-4
(a :b:c) = (1: -2: 1)
hence the given plane is
ax + by + cz = 0
x - 2y + z = 0
is the required equation of the plane
its normal is
i - 2j + k