Analytical mechanics for students of physics and engineering . equal c vanishes. Thereforetwo vectors of equal magnitude and opposite directions arethe negatives of each other. In other words, ,/•/„ „ the din<-Hon of a vector is reversed its sign is changed. (c) When a and b are at right angles to each other, as in Fig. 6c, = l- Therefore c2 = a2 + b2 and tan 0 = - • a 12. Difference of Two Vectors. — Subtraction is equivalent tothe addition of a negative quantity. Therefore, to subtractb from a we add — b to a. Thuswe have the following rule forsubtracting one vector from an-other. In orde
Analytical mechanics for students of physics and engineering . equal c vanishes. Thereforetwo vectors of equal magnitude and opposite directions arethe negatives of each other. In other words, ,/•/„ „ the din<-Hon of a vector is reversed its sign is changed. (c) When a and b are at right angles to each other, as in Fig. 6c, = l- Therefore c2 = a2 + b2 and tan 0 = - • a 12. Difference of Two Vectors. — Subtraction is equivalent tothe addition of a negative quantity. Therefore, to subtractb from a we add — b to a. Thuswe have the following rule forsubtracting one vector from an-other. In order to subtract one vectorfrom another reverse the one to besubtracted and add it to the othervector. It is evident from Fig. 7 thatthe sum and the difference of two vectors form the diagonalsof the parallelogram determined by them. ILLUSTRATIVE EXAMPLES. A particle is displaced 10 cm. X. 30° E., then Jo cm. E. Find theresulting displacement. Representing the displacements and their resultant by the vectors a, b, and c, Fig. 8, we obtain. ANALYTICAL MECHANICS c- = a- + 61 + 2 oo cos = (10 cm.)8 + (10 cm.)- + 2 X 10 cm. X 10 cm. cos (60°)= 300 tan 10 V3 cm.* h sin Y k b / ~> f S S a/ yf . ° ~-Jd ,X /.• 0 X a + b cos0 10 cm. sin (60°)10 cm.+ 10 cm. cos (00°)= J Vs. .: 6 = 30°. Therefore the resultani displacement is aboul 17.:; cm. along the direction X. G0° E. PROBLEMS. 1. A vector which points East has ;t length of 16 cm., and another vector which points Southeast is 25 cm. long. Find the direction andthe magnitude of their sum. 2. Find the direction and the magnitude of the difference of thevectors of the last problem. 3. The sum of two vectors is perpendicular to their difference. Showthat the vectors are equal in magnitude. 4. The sum and the difference of two vectors have equal that the vectors are at right angles to each other. 13. Resolution of Vectors into Compo-nents.— The projection of a vector upon aline is called t
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