Vector
components are added to determine the magnitude and direction of the resultant.
Calculations using trigonometric functions are the most accurate method for
making this determination.
EO 1.3ADD vectors using the following methods: c. Analytical
The graphic and components
addition methods of obtaining the resultant of several vectors described in the
previous chapters can be hard to use and time consuming. In addition, accuracy
is a function of the scale used in making the diagram and how carefully the
vectors are drawn. The analytical method can be simpler and far more accurate
than these previous methods.
Review of Mathematical Functions
In earlier mathematics lessons,
the Pythagorean Theorem was used to relate the lengths of the sides of right
triangles such as in Figure 22. The Pythagorean Theorem states that in any
right triangle, the square of the length of the hypotenuse equals the sum of
the squares of the lengths of the other two sides. This expression may be
written as given in Equation 24.
Figure 22 Right Triangle
Also, recall the three
trigonometric functions reviewed in an earlier chapter and shown in Figure 23.
The cosine will be used to solve for F_{x} The sine will be used to
solve for F_{ y}. Tangent will normally be used to solve for ,
although sine and cosine may also be used.
On a rectangular coordinate
system, the sine values of are positive (+) in quadrants I and II and
negative () in quadrants III and IV. The cosine values of are positive (+) in quadrants I and IV and
negative () in quadrants II and III. Tangent values are positive (+) in
quadrants I and III and negative () in quadrants II and IV.
Figure 23 Trigonometric
Functions
When mathematically solving for
tan ,
calculators will specify angles in quadrants I and IV only. Actual angles may
be in quadrants II and III. Each problem should be analyzed graphically to
report a realistic solution. Quadrant II and III angles may be obtained by
adding or subtracting 180° from the value calculated.
