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Addition of vectors given their magnitudes and directions
In this case, you must first translate the magnitudes and directions into
components, and the add the components.
Often the easiest way to add vectors is by making a scale drawing on a
piece of paper. This is known as graphical addition, as opposed to the
analytic techniques discussed previously.
Example
Question: Given the magnitudes and angles of the
.
r vectors
from San Diego to Los Angeles and from Los Angeles to Las
Vegas, find the magnitude and angle of the
.
r vector from San
Diego to Las Vegas.
Solution: Using a protractor and a ruler, we make a careful scale
drawing, as shown in the figure. A scale of 1 cm
.
100 km was
chosen for this solution. With a ruler, we measure the distance
from San Diego to Las Vegas to be 3.8 cm, which corresponds to
380 km. With a protractor, we measure the angle
.
to be 71
°
.
Even when we don’t intend to do an actual graphical calculation with a
ruler and protractor, it can be convenient to diagram the addition of vectors
in this way. With
.
r vectors, it intuitively makes sense to lay the vectors tip-
to-tail and draw the sum vector from the tail of the first vector to the tip of
the second vector. We can do the same when adding other vectors such as
force vectors.
Self-Check
How would you subtract vectors graphically.
Discussion Questions
A. If you’re doing graphical addition of vectors, does it matter which vector you
start with and which vector you start from the other vector’s tip.
B. If you add a vector with magnitude 1 to a vector of magnitude 2, what
magnitudes are possible for the vector sum.
C. Which of these examples of vector addition are correct, and which are
incorrect.
Los
Angeles
Las Vegas
San Diego
190 km
370 km
141
°
38
°
distance=.
.
=.
A
B
A+B
AB
A+B
A
B
A+B
B
A
A+
B
A
B
Vectors can be added graphically by
placing them tip to tail, and then
drawing a vector from the tail of the
first vector to the tip of the second
vector.
The difference A–B is equivalent to A+(–B), which can be calculated graphically by reversing B to form –B, and