Analytical mechanics for students of physics and engineering . ased upon the followinganalyiical method. 15. Analytical Method.—Expressing the given vectors and their resultanl in terms of their rectangular components, we have a= ar+ a„ + a„ b= bx+b„+b„ (1) r = rx + r„ + rzSubstituting from (1) in the vector equation r=a+b + c+ • • • (2) and collecting the terms we obtain r,+ rir+r. = (a,+ b,+ - • •)+(a„+b1/+ . .) +(a,+ b.+ - • •)• (3) But since the directions of the coordinate axes are indepen- •i the given vectors are in the same plane the parallelopiped reducesto a n ctangle. ADDITION AND R
Analytical mechanics for students of physics and engineering . ased upon the followinganalyiical method. 15. Analytical Method.—Expressing the given vectors and their resultanl in terms of their rectangular components, we have a= ar+ a„ + a„ b= bx+b„+b„ (1) r = rx + r„ + rzSubstituting from (1) in the vector equation r=a+b + c+ • • • (2) and collecting the terms we obtain r,+ rir+r. = (a,+ b,+ - • •)+(a„+b1/+ . .) +(a,+ b.+ - • •)• (3) But since the directions of the coordinate axes are indepen- •i the given vectors are in the same plane the parallelopiped reducesto a n ctangle. ADDITION AND RESOLUTION OF VECTORS 11 dent, the components of r along any one of the axes mustequal the sum of the corresponding components of th<vectors. Therefore (3) can be split into the following threeseparate equations. r*= a* + b,+ cx+ • • • ,1 r„= au+by+c„ + ? • •, I r, = a, + b, + c, + • • • . ] It was shown in § 11 that when two vectors are parallelthe algebraic sum of their magnitudes equals the magni-. Fig. 12. tude of their resultant. This result may be extended toany number of parallel vectors. Therefore we can put thevector equations of (4) into the following algebraic forms: rx = ax+bx+cx+ • • • ,1 ry=ay+by+cy+ • • •, rz = as + 6, + c, + • • • . |Equations (5) determine r through the following relations: vVs2+rv2+r,2, COS ct\ COS a3 = (7) rx — , COS c*2 = r / / where ah a2, and a3 are the angles r makes with the Multiplication and Division of a Vector by a a vector is multiplied or divided by a scalar the resultis a vector which has the same direction as the original vec-tor. If, in the equation b= ma. m be a scalar then b has thesame direction as a but its magnitude is m times thai of a. 12 ANALYTICAL MECHANICS ILLUSTRATIVE EXAMPLE. A man walks 3 miles X. 30° E., then one mile 1 ?:.. then 3 miles S. 45° E.,then 4 miles S., then one mile X. .30° W. Find his final position. Repr
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