By the 'angle between a vector
and a plane' we mean the angle between the vectorand a particular vector drawn in the plane. In the plane there are actually many vectors, and the particular vector we pick is the vector that makes the smallest angle with the vector
This is the angle shown as
in the diagram below.
If the equation of a plane
then the normal to the plane is the vector
and we can find the angle betweenand
using
Sinceis normal to the plane
so
Example: find the angle between the vector
and the plan
The normal vector to the plane is
so
So that
and