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A variety of mathematical operations can be performed
with and upon vectors. One such operation is the addition of
vectors. Two vectors can be added together to determine the
result (or resultant). This process of adding two or more
vectors has already been discussed in an
earlier unit. Recall in our discussion of Newton's laws
of motion, that the net force experienced by an
object was determined by computing the vector sum of all the
individual forces acting upon that object. That is the
net force was the
result (or resultant) of adding up
all the force vectors. During that unit, the rules for
summing vectors (such as force vectors) were kept relatively
simple. Observe the following summations of two force
vectors:

These rules for summing vectors were
applied to free-body
diagrams in order to determine the net force (i.e., the
vector sum of all the individual forces). Sample
applications are shown in the diagram below.

In this unit, the task of summing vectors
will be extended to more complicated cases in which the
vectors are directed in directions other than purely
vertical and horizontal directions. For example, a vector
directed up and to the right will be added to a vector
directed up and to the left. The vector sum will be
determined for the more complicated cases shown in the
diagrams below.

There are a variety
of methods for determining the magnitude and direction of
the result of adding two or more vectors. The two methods
which will be discussed in this lesson and used throughout
the entire unit are:

The
Pythagorean theorem is a useful method for determining the
result of adding two (and only two) vectors which make a
right angle to each other. The method is not applicable
for adding more than two vectors or for adding vectors which
are not at 90-degrees to each other. The Pythagorean
theorem is a mathematical equation which relates the length
of the sides of a right triangle to the length of the
hypotenuse of a right triangle.

To see how the method
works, consider the following problem:

Eric leaves the base camp and hikes
11 km, north and then hikes 11 km east. Determine
Eric's resulting displacement.

This problem asks to determine the result of adding two
displacement vectors which are at right angles to each
other. The result (or resultant) of walking 11 km north and
11 km east is a vector directed northeast as shown in the
diagram to the right. Since the northward displacement and
the eastward displacement are at right angles to each other,
the Pythagorean theorem can be used to determine the
resultant (i.e., the hypotenuse of the right triangle).

The result of adding 11 km, north plus 11
km, east is a vector with a magnitude of 15.6 km. Later,
the method of determining the direction of the vector will
be discussed.

Let's
test your understanding with the following two practice
problems. In each case, use the Pythagorean theorem to
determine the magnitude of the vector sum. When
finished, click the button to view the answer.

Using Trigonometry to Determine
a Vector's Direction

The direction of a
resultant vector can often be determined by use of
trigonometric functions. Most students recall the meaning of
the useful mnemonic SOH CAH TOA from their course in
trigonometry. SOH CAH TOA is a mnemonic which helps one
remember the meaning of the three common trigonometric
functions - sine, cosine, and tangent functions. These three
functions relate an acute angle in a right triangle to the
ratio of the lengths of two of the sides of the right
triangle. The sine
function relates the measure of an acute angle to
the ratio of the length of the side opposite the angle to
the length of the hypotenuse. The
cosine function relates
the measure of an acute angle to the ratio of the length of
the side adjacent the angle to the length of the hypotenuse.
The tangent function
relates the measure of an angle to the ratio of the length
of the side opposite the angle to the length of the side
adjacent to the angle. The three equations below summarize
these three functions in equation form.

These three trigonometric functions can be
applied to the hiker problem in order
to determine the direction of the hiker's overall
displacement. The process begins by the selection of one of
the two angles (other than the right angle) of the triangle.
Once the angle is selected, any of the three functions can
be used to find the measure of the angle. Write the function
and proceed with the proper algebraic steps to solve for the
measure of the angle. The work is shown below.

Once the measure of the angle is
determined, the direction of the vector can be found. In
this case the vector makes an angle of 45 degrees with due
East. Thus, the direction of this vector is written as 45
degrees. (Recall from earlier in
this lesson that the direction of a vector is the
counterclockwise angle of rotation which the vector makes
with due East.)

The measure of an angle as determined
through use of SOH CAH TOA is not always the
direction of the vector. The following vector addition
diagram is an example of such a situation. Observe that the
angle within the triangle is determined to be 26.6 degrees
using SOH CAH TOA. This angle is the southward angle of
rotation which the vector R makes with respect to West. Yet
the direction of the vector as expressed with the CCW
(counterclockwise from East) convention is 206.6
degrees.

Test
your understanding of the use of SOH CAH TOA to determine
the vector direction by trying the following two practice
problems. In each case, use SOH CAH TOA to determine the
direction of the resultant. When finished, click the button
to view the answer.

In the above problems, the magnitude and direction of the
sum of two vectors is determined using the Pythagorean
theorem and trigonometric methods (SOH CAH TOA). The
procedure is restricted to the addition of two vectors
which make right angles to each other. When the two
vectors which are to be added do not make right angles to
one another, or when there are more than two vectors to add
together, we will employ a method known as the head-to-tail
vector addition method. This method is described below.

Use
of Scaled Vector Diagrams to Determine a
Resultant

The magnitude and direction of the sum of two or more
vectors can also be determined by use of an accurately drawn
scaled vector diagram. Using a scaled diagram, the
head-to-tail method is
employed to determine the vector sum or resultant. A common
Physics lab involves a vector walk. Either using
centimeter-sized displacements upon a map or meter-sized
displacements in a large open area, a student makes several
consecutive displacements beginning from a designated
starting position. Suppose that you were given a map of your
local area and a set of 18 directions to follow. Starting at
home base, these 18 displacement vectors could be
added together in consecutive fashion to determine
the result of adding the set of 18 directions. Perhaps the
first vector is measured 5 cm, East. Where this measurement
ended, the next measurement would begin. The process would
be repeated for all 18 directions. Each time one measurement
ended, the next measurement would begin. In essence, you
would be using the head-to-tail method of vector
addition.

The head-to-tail method involves
drawing a vector to scale
on a sheet of paper beginning at a designated starting
position. Where the head of this first vector ends, the tail
of the second vector begins (thus, head-to-tail
method). The process is repeated for all vectors which are
being added. Once all the vectors have been added
head-to-tail, the resultant is then drawn from the tail of
the first vector to the head of the last vector; i.e., from
start to finish. Once the resultant is drawn, its length can
be measured and converted to real units using the
given scale. The direction of
the resultant can be determined by using a protractor and
measuring its counterclockwise angle of rotation from due
East.

A step-by-step method for applying the
head-to-tail method to determine the sum of two or more
vectors is given below.

Choose a scale and indicate it on a sheet of paper.
The best choice of scale is one which will result in a
diagram which is as large as possible, yet fits on the
sheet of paper.

Pick a starting location and draw the first vector
to scale in the indicated direction. Label the
magnitude and direction of the scale on the diagram
(e.g., SCALE: 1 cm = 20 m).

Starting from where the head of the first vector
ends, draw the second vector to scale in the
indicated direction. Label the magnitude and direction of
this vector on the diagram.

Repeat steps 2 and 3 for all vectors which are to be
added

Draw the resultant from the tail of the first vector
to the head of the last vector. Label this vector as
Resultant or simply R.

Using a ruler, measure the length of the resultant
and determine its magnitude by converting to real units
using the scale (4.4 cm x 20 m/1 cm = 88 m).

Measure the direction of the resultant using the
counterclockwise convention discussed earlier
in this lesson.

An example of the use of the head-to-tail
method is illustrated below. The problem involves the
addition of three vectors:

20 m, 45 deg. + 25 m, 300
deg. + 15 m, 210 deg.

SCALE: 1 cm = 5
m

The head-to-tail method is employed as
described above and the resultant is determined (drawn in
red). Its magnitude and direction is labeled on the
diagram.

SCALE: 1 cm = 5 m

Interestingly enough, the order in which three vectors
are added has no affect upon either the magnitude nor the
direction of the resultant. The resultant will still have
the same magnitude and direction. For example, consider the
addition of the same three vectors in a different order.

15 m, 210 deg. + 25 m, 300 deg. + 20 m, 45
deg.

SCALE: 1 cm = 5 m

When added together in this different order, these same
three vectors still produce a resultant with the same
magnitude and direction as before (22 m, 310 degrees). The
order in which vectors are added using the head-to-tail
method is insignificant.

SCALE: 1 cm = 5 m

Additional examples of vector addition using the
head-to-tail method are given on a
separate web page.