Addition of Three or More Right Angle Vectors
As our first example, consider the following
vector addition problem:
Example 1:
A
student drives his car 6.0 km, North before
making a right hand turn and driving 6.0 km to the East. Finally, the student
makes a left hand turn and travels another 2.0 km to the north. What is the
magnitude of the overall displacement of the student?
Like any problem in physics, a successful solution
begins with the development of a mental picture of the situation. The
construction of a diagram like that below often proves useful in the
visualization process.
When these three vectors are added together in
head-to-tail fashion, the resultant is a vector that extends from the tail of
the first vector (6.0 km, North, shown in red) to the arrowhead of the third
vector (2.0 km, North, shown in green). The head-to-tail vector addition
diagram is shown below.
As can be seen in the diagram, the resultant
vector (drawn in black) is not the hypotenuse of any right triangle - at least
not of any immediately obvious right triangle. But would it be possible to
force this resultant vector to be the hypotenuse of a right triangle? The
answer isYes! To do
so, the order in which the three vectors are added must be changed. The vectors
above were drawn in the order in which they were driven. The student drove
north, then east, and then north again. But if the three vectors are added in
the order 6.0 km, N + 2.0 km, N + 6.0 km, E, then the diagram will look like
this:
After rearranging the order in which the three
vectors are added, the resultant vector is now the hypotenuse of a right
triangle. The lengths of the perpendicular sides of the right triangle are 8.0
m, North (6.0 km + 2.0 km) and 6.0 km, East. The magnitude of the resultant vector (R) can be determined
using the Pythagorean theorem.
R2 =
(8.0 km)2 +
(6.0 km)2
R2 = 64.0 km2+
36.0 km2
R2 = 100.0 km2
R = SQRT (100.0 km2)
R = 10.0 km
(SQRT indicates square root)
In the first vector addition diagram above, the three
vectors were added in the order in which they are driven. In the second vector
addition diagram (immediately above), the order in which the vectors were added
was switched around. The size of the resultant was not affected by this change
in order. This illustrates an important point about adding vectors: the
resultant is independent by the order in which they are added. Adding vectors A + B + C gives the same
resultant as adding vectors B + A + C or even C + B + A. As long as all three vectors are included
with their specified magnitude and direction, the resultant will be the same.
This property of vectors is the key to the strategy used in the determination
of the answer to the above example problem. To further illustrate the strategy,
let's consider the vector addition situation described in Example 2 below.
Example 2:
Mac
and Tosh are doing the Vector
Walk Lab. Starting at the door of their physics classroom, they walk 2.0
meters, south. They make a right hand turn and walk 16.0 meters, west. They
turn right again and walk 24.0 meters, north. They then turn left and walk 36.0
meters, west. What is the magnitude of their overall displacement?
A graphical representation of the given problem will
help visualize what is happening. The diagram below depicts such a
representation.
When these four vectors are added together in
head-to-tail fashion, the resultant is a vector that extends from the tail of
the first vector (2.0 m, South, shown in red) to the arrowhead of the fourth
vector (36.0 m, West, shown in green). The head-to-tail vector addition diagram
is shown below.
The resultant vector (drawn in black and labeled R) in the
vector addition diagram above is not the hypotenuse of any immediately obvious
right trangle. But by
changing the order of addition of these four vectors, one can force this
resultant vector to be the hypotenuse of a right triangle. For instance, by
adding the vectors in the order of 2.0 m, S + 24.0 m, N + 16.0 m, W + 36.0 m.
W, the resultant becomes the hypotenuse of a right triangle. This is shown in
the vector addition diagram below.
With the vectors rearranged, the resultant is
now the hypotenuse of a right triangle that has two perpendicular sides with
lengths of 22.0 m, North and 52.0 m, West. The 22.0 m, North side is the result
of 2.0 m, South and 24.0 m, North added together. The 52.0 m, West side is the result of
16.0 m, West and
36.0 m, West added together. The magnitude of the resultant vector (R) can be
determined using the Pythagorean theorem.
R2 =
(22.0 m)2 +
(52.0 m)2
R2 = 484.0 m2 + 2704.0 m2
R2 = 3188.0 m2
R = SQRT (3188.0 m22)
R = 56.5 m
(SQRT indicates square root)
As can be seen in these two examples, the resultant of
the addition of three or more right angle vectors can be easily determined
using the Pythagorean theorem. Doing so involves the adding of the vectors in a different
order.