Hi, my name is Rosa Estela
and I am going to present this video
talking about linear algebra
and practicing with vectors.
First example is about how to normalize a vector.
Let's consider the vector v
with this components
and to normalize this vector,
we need to calculate the norm, and then the normalized vector
will be the vector divided by the norm.
The norm, remember, is the square root of the sum
of the squared components of the vector.
We do the operations and we have
that the norm is one half of root of 53.
So, the normalized vector of v
is this vector.
Another example
Important, about operations with vectors
is the linear combination of vectors let's consider
the vectors v_1, … ,v_n and the scalars c_1, …, c_n.
A linear combination of these vectors is
c_1v_1+ … +c_nv_n
For example, in R^2, the plane,
the vector (7,10) is a linear combination of the vectors
(1,2) and (2,2) because three times
the vector (1,2) plus two times the vector (2,2)
is the vector (7,10).
Another example of linear
combinations is this one.
The vector
(-27,-22,6)
In the three-dimensional space
is a linear combination of the vectors,
(1, -4 , 2) , (6, 2, 0)
and the vector (4, 2, 3) because three times
the vector (1, -4, 2)
-5 times
the vector (6, 2, 0)
is the vector.
The given vector.
This example is to see that
the scalar can be zero, like in this example.
Now let's consider an
example of scalar product
as a multiplication of vectors.
Remember that the scalar product of two vectors
is just a number and the vector product is a vector.
Then now we will have a number.
Let be the vectors
x = ( 1, 2, 3 ) and
y = ( 2, 3, 4 )
The scalar product of these two vectors
x · y is the product of the components
one times two plus
two times three plus
three times four.
That is 20.
And see that
with the other definition of the scalar product
we can have
the angle between these two vectors.
Thus, the cosines of the angle
is the scalar product 20
divided by the product of the norms
of these two vectors x and y.
And so, we have the angle
of these two vectors.
The last example
will be an example showing the orthogonality
of two vectors.
As we see in the picture, we consider the vectors
( 5, 3 ) and ( -3, 5 ).
You see in the picture that these vectors are orthogonal
because the angle between these two vectors
is pi
divided by two.
And let's show that the scalar
product is zero,
as we can see on the screen.
Thank you.