The preview activity presented us with a vector
and led us through the process of describing all the vectors orthogonal to
Notice that the set of scalar multiples of
describes a line
a 1-dimensional subspace of
We then described a second line consisting of all the vectors orthogonal to
Notice that every vector on this line is orthogonal to every vector on the line
We call this new line the
orthogonal complement of
and denote it by
The lines
and
are illustrated on the left of
Figure 6.2.2.
A typical example appears on the right of
Figure 6.2.2. Here we see a plane
a two-dimensional subspace of
and its orthogonal complement
which is a line in
Example 6.2.4.
If
is the line defined by
in
we will describe the orthogonal complement
the set of vectors orthogonal to
If is orthogonal to it must be orthogonal to so we have
We can describe the solutions to this equation parametrically as
Therefore, the orthogonal complement is a plane, a two-dimensional subspace of spanned by the vectors and
Example 6.2.5.
Suppose that is the -dimensional subspace of with basis
We will give a description of the orthogonal complement
If is in we know that is orthogonal to both and Therefore,
In other words, where
The solutions may be described parametrically as
The distributive property of dot products implies that any vector that is orthogonal to both and is also orthogonal to any linear combination of and since
Therefore, is a -dimensional subspace of with basis
One may check that the vectors and are each orthogonal to both and