We know that

The Cartesian equation of a line through a point (x₁, y₁, z₁) and having direction cosines l, m, n is .

Comparing this standard form with the given equation, we get

x₁ = 5, y₁ = -4, z₁ = 6 and l = 3, m = 7, n = 2

⇒ The point through which the line passes has the position vector and the vector parallel to the line is given by .

Now, ∵ Vector equation of a line that passes through a given point whose position vector is and parallel to a given vector is .

∴ The vector equation of the required line is:

⇒