Components along the same axis, say the
x -axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the
y -axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of
are known, its magnitude and direction can be found.
Step 3.To get the magnitude
of the resultant, use the Pythagorean theorem:
Step 4.To get the direction of the resultant:
The following example illustrates this technique for adding vectors using perpendicular components.
Adding vectors using analytical methods
Add the vector
to the vector
shown in
[link] , using perpendicular components along the
x - and
y -axes. The
x - and
y -axes are along the east–west and north–south directions, respectively. Vector
represents the first leg of a walk in which a person walks
in a direction
north of east. Vector
represents the second leg, a displacement of
in a direction
north of east.
Vector
has magnitude
and direction
north of the
x -axis. Vector
has magnitude
and direction
north of the
x -axis. You can use analytical methods to determine the magnitude and direction of
.
Strategy
The components of
and
along the
x - and
y -axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.
Solution
Following the method outlined above, we first find the components of
and
along the
x - and
y -axes. Note that
,
,
, and
.
We find the
x -components by using
, which gives
and
Similarly, the
y -components are found using
:
and
The
x - and
y -components of the resultant are thus
and
Now we can find the magnitude of the resultant by using the Pythagorean theorem:
so that
Finally, we find the direction of the resultant:
Thus,
Using analytical methods, we see that the magnitude of
is
and its direction is
north of east.
Discussion
This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector.
Subtraction of vectors is accomplished by the addition of a negative vector. That is,
. Thus,
the method for the subtraction of vectors using perpendicular components is identical to that for addition . The components of
are the negatives of the components of
. The
x - and
y -components of the resultant
are thus
Receive real-time job alerts and never miss the right job again
Source:
OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
Google Play and the Google Play logo are trademarks of Google Inc.
Notification Switch
Would you like to follow the 'Sample chapters: openstax college physics for ap® courses' conversation and receive update notifications?