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
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
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.
Let be a point not on the line in .
- Consider the function
Prove is a quadratic polynomial in and that there is a unique value of , say , at which this polynomial takes on its minimum.
- Prove that is orthogonal to .
- Proof. First, we compute
But, are all just real scalars (by definition of the dot product); therefore,
for scalars given by
which implies has a unique minimum at
- Proof. We compute the dot product,
Then we have