The vector equation of a line passing through two points with position vectors & is

⇒

The position vector of point A(5,1,6) is given as -

.

The position vector of point B(3,4,1) is given as -

⇒

⇒

∴ ...(1)

Let the coordinates of the point where the line crosses the ZX plane be (0,y)

So, …(2)

Since point lies in line, it will satisfy its equation,

Putting (2) in (1)

Two vectors are equal if their corresponding components are equal

So,

x = 5 - 2λ …(3)

0 = 1 + 3λ …(4)

and, z = 6 - 5λ …(5)

From equation (4), we get -

λ = -1/3

Substitute the value of λ in equation (3) and (5), we get -

x = 5 - 2λ = 5 - 2 × (-1/3) = 5 + (2/3) = 17/3

and

z = 6 - 5λ = 6 - 5 × (-1/3) = 6 + (5/3) = 23/3

Therefore, the coordinates of the required point is

(17/3, 0, 23/3).