Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes and

We know that the equation of line passing through (1,2,3) is given by

……(1)

We know that line is parallel to plane a₂x + b₂y + c₂z + d₂ = 0 if a₁a₂ + b₁b₂ + c₁c₂ = 0 ……(2)

Here, line (1) is parallel to plane,

x – y + 2z = 5

So,

a×1 + b× – 1 + c×2 = 0

⇒ a – b + 2c = 0 ……(3)

Also, line (1) is parallel to plane,

3x + y + z = 6

So,

a×3 + b×1 + c×1 = 0

⇒ 3a + b + c = 0 ……(4)

Solving equation (3) and (4) by cross multiplication we have,

⇒

⇒

∴ a = – 3k, b = 5k and c = 4k

Putting the value in equation (1)

Multiplying by k we have

The required equation is