Welcome to a lesson on orthogonal
and orthonormal sets of vectors.
A set of vectors is called orthogonal
if all the vectors in the set
are perpendicular to each other.
A set of nonzero vectors containing the vectors,
u sub one through u sub m is orthogonal
if the dot product of vector u sub i
and vector u sub j equals zero whenever i doesn't equal j.
An orthonormal set of vectors is linearly independent.
A set of vectors is called orthonormal
if the orthogonal set is also a set of unit vectors.
And therefore, a set of nonzero vectors
containing the vectors u sub one through u sub m
is orthonormal if the dot product
of vector u sub i and vector u sub j equals zero
whenever i doesn't equal j,
and vector u sub i dotted with itself
is equal to one for all values of i from one to m.
Whenever we dot a unit vector with itself,
the result is always one.
As an example, the standard basis vectors
or standard coordinate vectors in Rn
always form an orthogonal and orthonormal set.
Let's consider R3,
where the standard basis vectors
are the vectors one, zero, zero,
or vector e sub one,
the vector zero, one, zero
or the vector e sub two,
and the vector zero, zero, one,
or the vector e sub three.
This is an orthogonal set,
as well as an orthonormal set of vectors.
Notice how the dot product of vector e sub one
and vector e sub two equals zero,
as well as the dot product
of vector e sub one and vector e sub three,
and as well as the dot product
of vector e sub two and the vector e sub three.
So because all the dot products are equal to zero,
we know this set is orthogonal.
And also notice how,
if we take each vector in the set
and dot it with itself,
the result is always positive one,
which indicates the set
is also an orthonormal set of vectors,
meaning all the vectors are unit vectors,
as well as orthogonal to each other.
On the left,
we have the graph of the standard basis vectors in R3.
And we can see all the vectors are unit vectors,
and they are also perpendicular or orthogonal to each other.
The set of vectors v sub one through v sub m
is an orthogonal set of vectors.
Then if we take each of the vectors
and divide by their magnitudes,
forming unit vectors,
the new set is an orthonormal set of vectors.
Let's take a look at some examples.
We're asked to determine if the sets
of vectors are orthogonal.
To do this, we need to check the dot products.
For the first set, because we only have two vectors,
we simply need to check the dot product
to see if it's equal to zero.
So if we call the first factor vector u sub one
and the second vector u sub two,
vector u sub one dotted with vector u sub two
is equal to two times three
plus negative three times two,
which is six plus negative six,
which equals zero.
So because the dot product is zero,
the set is an orthogonal set of vectors.
In the next set, we have three vectors,
and therefore we'll have to find vector u sub one
dotted with vector u sub two,
vector u sub one dotted with vector u sub three,
as well as vector u sub two
dotted with vector u sub three.
If all the dot products are zero,
then the set is an orthogonal set of vectors.
So let's start with vector u sub one
dotted with vector u sub two,
which gives us one times zero,
plus zero times one
plus one times zero,
which is zero.
We now have find the dot product of vector u sub one
and vector u sub three,
which is one times negative one
plus zero times zero,
plus one times one,
which is negative one plus one,
which is zero.
And now let's check the dot product of vector u sub two
and vector u sub three,
which is zero times negative one
plus one times zero
plus zero times one
which is also zero.
So because all the dot products are zero,
the set of vectors is orthogonal.
Let's take a look at one more set of vectors.
Again, because we have three vectors in the set,
we'll have to determine three dot products.
Again, let's begin with vector u sub one
dotted with vector u sub two
which is the first vector dotted with the second vector.
This gives us two times negative one
plus three times one
plus one times negative one,
which is negative two plus three
plus negative one, which is zero.
And now let's check the dot product
of vector u sub one and vector u sub three,
which gives us two times two
plus three times two
plus one times zero,
which is four plus six or positive 10.
So we can stop here because this dot product is not zero.
This is not an orthogonal set of vectors.
Let's go ahead and just check the dot product
of vector u sub two and vector u sub three.
Notice how this dot product is zero,
but again, as long as we have one dot product
that's not zero, the set of vectors is not orthogonal.
So again, this set is not orthogonal.
And then for our last example,
we are told the given set of vectors is orthogonal.
We're asked to find an orthonormal set
of vectors with the same span.
So even though we're told it's orthogonal,
let's go ahead and check the dot products anyway.
Notice all the dot products are zero,
and now to find an orthonormal set of vectors
with the same span,
we need to find the magnitude of each of these three vectors
and then divide each of the components of the vectors
by the magnitude of the vector.
So let's begin with the first vector
which we'll call vector v sub one
and find the magnitude,
which is equal to these square root of the sum
of the square of three,
plus the square of zero,
plus the square of two,
which equals the square root of 13.
And now let's find the magnitude of the second vector
which we'll call vector v sub two,
which is equal to the square root of the sum
of the square of zero plus the square of five,
plus the square of zero,
which is equal to the square root of the square of five,
which equals five.
And then finally for the third vector,
the magnitude is equal to the square root
of the sum of the square of negative two,
the square of zero and the square of three,
which equals the square root of 13.
And therefore, an orthonormal set of vectors
with the same span as the given set of vectors
is equal to the set shown here,
where notice how we divided each component
of the first vector by the square root of 13,
we divided every component of the second vector by five,
and we divided the components of the third vector
by the square root of 13.
So for this set of vectors,
all the vectors are orthogonal or perpendicular
and they're also unit vectors which is why we call
this an orthonormal set of vectors.
I hope you found this helpful.