Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:

and .

The vector equation of a line passing through a point with position vector and parallel to a vector is

⇒

Given, the line passes through (1, 2, -4)

So,

Given, line is parallel to both planes

∴ Line is perpendicular to normal of both planes.

i.e is perpendicular to normal of both planes.

we know that -

is perpendicular to both &

So, is cross product of normals of planes

and

Required Normal

Thus,

Now, putting the value of & in formula

Therefore, the equation of the line is