Vector Addition
Tuesday, December 10, 2013
4:32 PM
Slide
Notes
Let's start out with some examples of when
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Let's start out with some examples of when
vector addition is useful. A boat in a river will not
only be influences by the direction the boat is
pointing but it will be influenced by the current of
the water it is in. The same thing goes for the
direction a plane is pointing but it having a cross
wind that is pushing it in a different direction
somewhat.
You can think of vector addition as almost finding
an average position between the two original
Let's try an example. These take many steps, so it
is helpful to organize you work area up front. You
need a chart for finding your x and y. A sketch
area to see what your vectors look like. You will
not need to be exact or draw these to scale, but
you do need to have enough to spot the quadrant
that they fall in.
We can estimate a bit where our final vector will
fall. In this case, since the angels are the same in
each triangle but the top vector is a little longer
than the bottom, I estimate that the final vector
will fall in quadrant 1.
Also create an area just to remind you of the trig
relationships. That area at the bottom will also be
where you will work out the resulting vector's
magnitude and angle.
Begin by adding in the information in to the chart
about each vector that is being added and sketch
it in to the coordinate sketch area. It may be
tempting to skip this step, but it is often very
important so that you get the real angle inside the
triangle. Notice that vector B is 290 degrees. That
will work out to actually be a triangle of 60
degrees, since the triangles must have a leg that
falls on the x axis and they must be right triangles.
Identify what side is the x and y and correlate that
to the opposite or adjacent leg indications for trig.
For Vector A, the x will be adjacent leg, so we
need the trig function with the hypotenuse and
the adjacent leg. That is cos. We will solve for the
adjacent leg by multiplying the hypotenuse and
the cos of 60. We get 1.6. Vector A is in the first
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the cos of 60. We get 1.6. Vector A is in the first
quadrant so the x and the y will be positive.
The y will be opposite and hypotenuse which is
the sin function. We will multiply the hypotenuse
to the sin of 60 degrees and get 2.7.
We do the same for vector B, but note that vector
B is in quadrant 4. That means that the y will be a
negative.
Now we add the x of vector A and vector B. That
gives us 2.1.
Then we will add the y-component of each vector
to each other and get 1.4.
Now that we have the x and the y values, we will
sketch out the vector and the resulting right
triangle of the resulting vector. Our estimate that
the resulting vector will fall in quadrant 1 is
confirmed.
Now that we have the sides, we will be able to
use the Pythagorean Theorem to find the length
of the hypotenuse which is part of the final
answer. We get 2.5. Our units in this example are
m/s.
The last part is to get out angle. When we solve
for the angle in trig, we use the inverse trig
functions. Though we did calculate the
hypotenuse and could use it, it is one level deeper
into the calculations and that means a higher risk
we made an error, so we will use the opposite
and the adjacent legs. That means we need the
inverse tangent function. The inverse tangent of
1.4 divided by 2.1 is 34 degrees.
Our final answer in polar form is 2.5 m/s at an
angle of 34 degrees.
Beginning of Try It Section
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Beginning of Try It Section
Find x and y
Find magnitude (hypotenuse)
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Find magnitude (hypotenuse)
Find Angle
Congratulations
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Congratulations
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