Consider a plane with the Cartesian equation
Find each of the following:
- A normal vector to the plane with length 1.
- The
intercepts of the plane.
- The distance from the plane to the origin.
- The point on the plane,
, nearest to the origin.
- From the Cartesian equation we know that a normal vector to the plane is
. Making this of unit length, we have a unit normal
given by
- Since the Cartesian equation gives us
we have the
-intercept is
, the
-intercept is
and the
-intercept is
.
- The distance from the plane to the origin is given by
- The point
on the plane which is nearest to the origin is in the direction of the normal vector, and at a distance of
, therefore,
d) is incorrect here and correct in the book. The catch is that (PN) is negative, so the normal vector point in the opposite direction from the vector P on the plane, so we should negate it’s scaled version in order for that point to be on the plane.