Find the vector and Cartesian equations of the line passing through the point (2,1,3) and perpendicular to the lines x=y/2=z/3 and x=y/2=z/5.

Let the required line be parallel to the vector given by, = b₁i + b₂j + b₃k
The position vector of the point (2,1,3) and parallel to vector is
r = a + λb
⇒ r = (2i + j + 3k) + λ(b₁i + b₂j + b₃k) — (1)
The equation of the lines are
x = y/2 = z/3 — (2)
x = y/2 = z/5 —- (3)
Line (1) and (2) are ⊥ to each other.
∴ b₁ + 2b₂ + 3b₃ = 0 — (4)
Also, Line (1) and (3) are ⊥ to each other.
∴ b₁ + 2b₂ + 5b₃ = 0 — (5)
From equation (4) and (5), we obtain
b₁/2(5-3) = b₂/5-3 = b₃/2-2
b₁/4 = b₂/2 = b₃/-2
b₁/2 = -b₂/1 = b₃/-4
⇒b₁/2 = -b₂/1 = b₃/4
Therefore direction ratios of are 2, -1, 4.
b = 2i - j + 4k
Substituting b = 2i - j + 4k in equation (1), we obtain
r = (2i + j + 3k) + λ(2i - j + 4k)
This is the equation of the required line.