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.
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,
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
and
and the rest of the method outlined above is identical to that for addition. (See
[link] .)
Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module,
Projectile Motion , is one of many in which using perpendicular components helps make the picture clear and simplifies the physics.
Questions & Answers
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A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon