803 8.3 Geometrically Defined Vectors and Applications The Equilibrant The previous example showed a method for finding the resultant of two vectors. Sometimes it is necessary to find a vector that will counterbalance the resultant. This opposite vector is the equilibrant. That is, the equilibrant of vector u is the vector -u. EXAMPLE 1 Finding the Magnitude of a Resultant Two forces of 15 and 22 newtons act on a point in the plane. (A newton is a unit of force that equals 0.225 lb.) If the angle between the forces is 100°, find the magnitude of the resultant force. SOLUTION As shown in Figure 26, a parallelogram that has the forces as adjacent sides can be formed. The angles of the parallelogram adjacent to angle P measure 80° because adjacent angles of a parallelogram are supplementary. Opposite sides of the parallelogram are equal in length. The resultant force divides the parallelogram into two triangles. Use the law of cosines with either triangle. v 2 = 152 + 222 - 211521222 cos 80° Law of cosines v 2 ≈225 + 484 - 115 v 2 ≈594 Add and subtract. v ≈24 To the nearest unit, the magnitude of the resultant force is 24 newtons. S Now Try Exercise 27. v 808 808 100° 15 22 15 22 P Q S R Figure 26 Evaluate powers and cos 80°. Multiply. Take square roots and choose the positive root. EXAMPLE 2 Finding the Magnitude and Direction of an Equilibrant Find the magnitude of the equilibrant of forces of 48 newtons and 60 newtons acting on a point A, if the angle between the forces is 50°. Then find the angle between the equilibrant and the 48-newton force. SOLUTION As shown in Figure 27, the equilibrant is -v. v –v A a B C 48 60 48 60 130° 50° Figure 27 The magnitude of v, and hence of -v, is found using triangle ABC and the law of cosines. v 2 = 482 + 602 - 214821602 cos 130° Law of cosines v 2 ≈9606.5 Use a calculator. v ≈98 To the nearest unit, the magnitude is 98 newtons. The required angle, labeled a in Figure 27, can be found by subtracting angle CAB from 180°. Use the law of sines to find angle CAB. Square root property; Give two significant digits.
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