Home » Blog » Determine whether a line is parallel to given planes

Determine whether a line is parallel to given planes

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.

1. The plane through the point and spanned by and .
2. The plane through the points .
3. The plane determined by the Cartesian equation .

1. 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.

2. 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 .

3. 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.