The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. The direction of a resultant vector can often be determined by use of trigonometric functions.
Using Trigonometry to Determine a Vector's Direction When finished, click the button to view the answer. In each case, use the Pythagorean theorem to determine the magnitude of the vector sum. Let's test your understanding with the following two practice problems. Later, the method of determining the direction of the vector will be discussed.
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The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle). The result (or resultant) of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other. To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other.
The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other.
There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The vector sum will be determined for the more complicated cases shown in the diagrams below. For example, a vector directed up and to the right will be added to a vector directed up and to the left. In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions.
Sample applications are shown in the diagram below. These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Observe the following summations of two force vectors: During that unit, the rules for summing vectors (such as force vectors) were kept relatively simple. That is the net force was the result (or resultant) of adding up all the force vectors. Recall in our discussion of Newton's laws of motion, that the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. This process of adding two or more vectors has already been discussed in an earlier unit. Two vectors can be added together to determine the result (or resultant). One such operation is the addition of vectors. A variety of mathematical operations can be performed with and upon vectors.
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