By Robert J. Nikolai

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**Extra resources for Bioengineering Analysis of Orthodontic Mechanics**

**Sample text**

Vector addition. The relationship among the displacement vectors of points in Figure 1-5. 20 Bioengineering Analysis of Orthodontic Mechanics cent sides of a parallelogram. The third and fourth sides of the parallelo- gram are added (dashed) to complete the figure and the desired vector, the absolute displacement of point D, is coincident with a diagonal. The length of the resultant vector is measured and multiplied by the scale factor to obtain the magnitude of the displacement of point D. Because the two vectors summed in Figure 1-13 are nearly parallel and have comparable senses, the result in this example is almost as if the component magnitudes were added.

Hand-held, electronic calculators and digital computers will yield numerical solutions to six or more figures, but no more digits should appear in the answer than can be justified. The evaluation of the accuracy of mathematical procedures is a complex matter and will not be investigated here. It is perhaps sufficient to indicate that most input data in orthodontic bioengineering problems are accurate to no more than three significant figures and, therefore, no more than three significant figures should be recorded in the solutions obtained.

The force magnitudes are represented in the lengths of the vectors according to the given scale; because it is twice the magnitude, the length of the headgear force vector is twice that of the elastic force. The given forces intersect at point 0, the center of the bracket in the buccal view, and their vector representations form adjacent sides of a parallelogram. The dashed lines are added to complete the geometric figure. The vector representing the resultant is drawn as the diagonal of the parallelogram; the "tail" of the resultant vector coincides with the point where the "tails" of the given vectors meet.