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Mathematical Expression Editor
Linear Combinations of Vectors
When studying vectors, the two main operations we have learned about are vector
addition and scalar multiplication. Both are involved in the important concept of a
linear combination of vectors.
A vector is said to be a linear combination of vectors if
for some scalars .
For example, is a linear combination of , and because
In this section we will focus on vectors in and .
Visualizing Linear Combinations in and
Answer the questions below using the GeoGebra interactive. To use the interactive,
you can
(a)
Change vectors and by dragging the tips of these vectors.
(b)
Change the coefficients and of the linear combination by using sliders.
(a)
Let and . Find and such that
(b)
Let and . Find and such that
(c)
Use the same vectors and as in the previous part. Do you think it is possible
to express any vector in as a linear combination of and ? Yes, No
(d)
Let and . Do you think it is possible to express any vector in as a linear
combination of and ? Yes, No
Visualizing linear combinations of vectors in is more difficult than doing so in . The
following GeoGebra interactive will help you do this.
To use the interactive, define vectors , and . Use sliders to change the coefficients
and of the linear combination. The linear combination is shown as the pink vector
along the diagonal of the parallelepiped.
RIGHT-CLICK and DRAG the left panel to rotate the image.
Geometry of Linear Combinations
Use geometry to express as a linear combination of and .
We are looking for to be the diagonal of a parallelogram determined by scalar
multiples of and .
Because a scalar multiple of a vector can point in the same direction as the vector or
in the opposite direction, we will start by drawing straight lines determined by the
two vectors.
The two lines that we drew will contain the sides of the parallelogram we are looking
for. To find the other two sides we will draw lines parallel to and through the head
of vector .
Now the parallelogram is clearly visible.
The last remaining task is to identify the sides of the parallelogram as scalar
multiples of and . We do this by identifying vectors and as the vectors that
determine the parallelogram.
Observe that vector is half the length of and points in the opposite direction, while
the vector is the same length as and also points in the opposite direction.
Now we can write as a linear combination of and as follows
The method we used in Example ex:lincombparallelogrammethod to express the given vector as a linear
combination of two other vectors is sufficiently useful that we summarize the
steps.
Given two non-collinear vectors and in , and a vector , we can express as a linear
combination of and as follows:
(a)
Draw lines and determined by and , respectively.
(b)
Through the head of vector , draw lines and , parallel to and , respectively.
(c)
Let be the point of intersection of and .
(d)
Let be the point of intersection of and .
(e)
Let denote the origin. Then and for some scalars and .
(f)
We have .
This GeoGebra interactive will allow you to go through the steps given in Procedure
pro:lincombgeo for a combination of vectors of your choice. To use the interactive
(a)
Enter components of vectors and .
(b)
Enter components of vector that you want to express as a linear
combination of and .
(c)
Use the navigation bar to go through the steps of Procedure pro:lincombgeo
From Geometry to Algebra of Linear Combinations
One of the stipulations in Procedure pro:lincombgeo is that vectors and should be non-collinear.
You can use the interactive in Exploration exp:proc4 to investigate what happens when and
are collinear. The following example examines what happens from a geometric as well
as an algebraic standpoint.
Can the vector be written as a linear combination of vectors and ?
We will start
with a geometric approach.
Observe that and are scalar multiples of each other and lie on the same
line.
A linear combination of and has the form:
This shows that all linear combinations of and will be scalar multiples of , and
therefore lie on the same line as . Since does not lie on the line determined by it
cannot be expressed as a linear combination of and .
We can also address this question algebraically. To express as a linear combination of
and , we need to solve the equation.
This gives us a system of equations
When you try to solve this system, you will find that the system is inconsistent. Thus,
cannot be written as a linear combination of and .
We know that there is no way to express as a linear combination of vectors and .
What would happen if we tried to apply Procedure pro:lincombgeo to these vectors? You can use the
GeoGebra interactive in Exploration exp:proc4 to find out.
Express as a linear combination of and . Interpret your results geometrically.
We
need to find scalars and such that
This amounts to solving a system of linear equations
Use your favorite method to solve this system. (Hint: adding the second equation to
the first will work well for this system.) You will find that and . Now we can write
as a linear combination of and as follows:
Geometrically speaking, this means that the vector is the diagonal of the
parallelogram determined by and . The original vectors and are shown below
together with the parallelogram and its diagonal.
If possible, express as a linear combination of and .
We are looking for coefficients
and such that
This translates into a system of equations
Solve this system for and , and enter your answers below:
We conclude that is a linear combination of and , and write:
Set up a system of equations that can be used to express as a linear combination of ,
, and , or to determine that such a combination does not exist. Do not solve the
system.
We are looking for , , and such that
This translates into the following system of equations:
Practice Problems
Solve a system of linear equations to express as a linear combination of and
.
System of linear equations:
Values of and :
Linear Combination:
Use Procedure pro:lincombgeo to express as a linear combination of and .
Linear Combination:
Use two different approaches (algebraic and geometric) to explain why the vector
cannot be expressed as a linear combination of vectors and .
We have seen Procedure pro:lincombgeo applied to vectors in . The same process can, in certain
cases be applied to vectors in . In both parts of this problem you will be asked to
follow the steps in Procedure pro:lincombgeo to express one vector as a linear combination of two
given vectors. Then you will be asked to identify the condition which makes it
possible to do so.
The following GeoGebra interactive shows vectors , and . RIGHT-CLICK and
DRAG to rotate the image.
(a)
Can be expressed as a linear combination of and ?
No, because is not
between and .Yes, because all three vectors are in the same plane.Yes,
because all three vectors are in the same plane, AND and are not collinear.
(b)
Use the navigation bar at the bottom of the interactive window to view
construction steps of Procedure pro:lincombgeo applied to vectors , and . (Right-click and
drag to rotate the image.) Use the final image to express as a linear
combination of (blue) and (red).
The following GeoGebra interactive shows vectors , , and . RIGHT-CLICK and
DRAG to rotate the image. Use geometry to explain why cannot be expressed as a
linear combination of and .
We can also show that is not a linear combination of and algebraically by
attempting to solve a system of equations corresponding to
Set up the system of equations