We say that a line in the direction of a vector
is parallel to a plane
if
is parallel to
. Consider the line
through the point
and parallel to the vector
. Determine whether
is parallel to the following planes.
- The plane through the point
and spanned by
and
.
- The plane through the points
.
- The plane determined by the Cartesian equation
.
- This asks if
is in the span of
, i.e., does there exist
such that
From the second equation we have
. Then, from the first,
which implies
But then,
Thus, there is no solution, so the line is not parallel to the plane.
- The plane through the points
is the set of points
For
to be in the span of
we must have
such that
From the first equation we have
. Then from the second we have
which implies
But then,
Hence,
is not parallel to
.
- The plane with Cartesian equation
is the set of points
The points
are all in
. So,
Thus, we ask if
is in the span of
. This requires that there exist
such that
But, this fails since the second and third equations implies
and
. But then
Hence, this line is not parallel to the plane.