Find the position vector of the foot of the perpendicular and the perpendicular distance from the point P with position vector to the plane Also, find the image of P in the plane.

Let the position vector of P be so that and M be the image of P in the plane.

In addition, let Q be the foot of the perpendicular from P on to the given plane so that Q is the midpoint of PM.

Direction ratios of PM are proportional to 2, 1, 3 as PM is normal to the plane and parallel to.

Recall the vector equation of the line passing through the point with position vector and parallel to vector is given by

Here, and

Hence, the equation of PM is

Let the position vector of M be. As M is a point on this line, for some scalar α, we have

Now, let us find the position vector of Q, the midpoint of PM.

Let this be.

Using the midpoint formula, we have

This point lies on the given plane, which means this point satisfies the plane equation.

We have the image

Foot of the perpendicular

Using the distance formula, we have

Thus, the position vector of the image of the given point is and that of the foot of perpendicular is. Also, the length of this perpendicular is units.