Hi,

I need your help to get me started. Im totally lost.

Find the perpendicular distance of the plane 5x + 2y - z = -22 from origin O by first finding the co-ordinates of the point P on the plane such that OP is perpendicular to the given plane.

Thank You.

The first thing you're looking for is an equation of a line normal to the plane. Furthermore, we want to make sure the line passes through the origin.

This first step is pretty easy: first find a vector normal to the plane. This part is totally trivial; the unit vector coefficients are just the coefficients of the x, y, and z coordinates of the plane, n=5i + 2j-k. Next, we can write the parametric equation of a line parallel to that unit vector. It's equation would be





where the point is a point through which we want the line to pass. In this case, that point is the origin, so the parametric equation is even simpler:





At the point where this line intersects the plane, the equations of the plane and the line will all be simultaneously satisfied. That means we can plug in the values of x, y, and z for the line (which will be in terms of the parameter t) into the equation of the plane:



Solving this for the value of t where the two intersect, we get



Now that we know the value of the parameter t at the point of intersection, we can convert that back into Cartesian coordinates by plugging it into our equations for the line:





So our point of intersection is



There! We now have the point of intersection between the plane and a line normal to it passing through the origin. The last step is to simply find the distance between those two points, which I assume you can do.

Hope that helps.