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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.