Find the equation of the line in vector and in cartesian form that passes through the point with position vector and is in the direction

We know that

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

Here, and

⇒ The vector equation of the required line is:

⇒

Also, we know that

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

Also, we know that if the direction ratios of the line are a, b, c, then

⇒

⇒ The Cartesian equation of a line through a point (x₁, y₁, z₁) and having direction ratios a, b, c is .

Here, x₁ = 2, y₁ = -1, z₁ = 4 and a = 1, b = 2, c = -1

⇒ The Cartesian equation of the required line is:

⇒