Greenwood press

154
VECTORS

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Street. Go southwest eight blocks on Maple.” Applications of vectors appear in
physics, navigation, computer graphics, airplane design, and statistics.
In the illustration below, Sarah and James are pulling a wagon. Sarah pulls
with 25 pounds of force, and James with 15 pounds. They are pulling at an angle
of 20°. What is the net direction and force on the wagon? The James vector 
j is
drawn along the 
x-axis of the grid. The three units represent 15 pounds of force.
The vector for Sarah, 
s, is 5 units long, representing 25 pounds of force. It is
drawn 20° from 
j. The parallelogram law in physics states that the resultant vec-
tor
r is the diagonal of the parallelogram that is formed with sides parallel to the
vectors. (See Quadrilaterals.) The obtuse angle in the triangle is formed by two
sides and the diagonal is 160°. By the law of cosines (see Triangle Trigonometry),
r
2
= s
2
+ j
2
− 2sj cos θ
r
2
= 25
2
+ 15
2
− 2(25)(15) cos 160°.
Solving for 
r yields a force of 39.43 pounds. Using the law of sines computes the
angle between 
r and  j to be about 12.5°. Because they are pulling at an angle,
the forces don’t add to the total possible for Sarah and James (40 pounds of
force), but they come close. 
The same kind of analysis applies to paths of airplanes. The next figure
shows an airplane pointed due northeast at 400 miles per hour. Its vector 
a is
drawn 45° clockwise from north. A 90-mile-per-hour wind is blowing 10° south
of east. The wind vector 
w is shown at 10° clockwise from east. The angle
between the plane vector and the wind vector is 55°. The plane will be blown
somewhat off course. What is its direction and ground speed? The situation
depicted indicates directions in degrees according to navigation conventions.
Complete the resultant vector 
r and compute its length and direction. From the