5 6.] TRANSLATIONS. 29 



(8) Resolve a vector a into two components making with the vector 

 angles of 30 and 45 on opposite sides. 



(9) Steering his boat directly across a river whose current is due 

 west, a man arrives on the opposite bank at a point from which the 

 starting-point bears S.E. ; the width of the river being 1200 feet, how 

 far has he rowed ? What is the absolute, and what the relative, displace- 

 ment of the boat ? 



(10) Assuming a raindrop to fall 25 feet in a second in a vertical 

 direction, find in what direction it appears to be falling to a man : (a) 

 walking at the rate of 5 feet per second, (b) driving at the rate of 10 

 feet per second, (r) riding on a bicycle at 25 feet per second, (d) in a 

 railroad car running 60 feet per second. 



(n) Find in magnitude and direction the resultant of 8 translations 

 of i, 2, 3, 4, 5, 6, 7, 8 feet, respectively, each component making an 

 angle of 45 with the preceding one : (a) graphically, (b) analytically. 



(12) If a, b, c are three vectors whose geometric sum is o, prove* 

 that a/sin (be) =^/sin (ca) =r/sin (a-6>). 



(13) Find the resultant of two translations represented in magnitude 

 and direction by two rectangular chords of a circle drawn from a point 

 on its circumference. 



(14) From a point C in the plane of a circle whose centre is O, 

 draw two lines at right angles to each other so as to intersect the circle 

 in A y A' and B, B\ respectively. Show that the resultant of the four 

 vectors CA, CA', CB, CB< is equal to twice CO. 



(15) Prove that the geometric sum of two vectors P^P^ PoP 2 issuing 

 from the same point P passes through the middle point G of P\P$ and 

 has a length = 2 P G. 



(16) Prove that the geometric sum of two vectors P$P\ and /o/a is 

 equal to ( -f i)P G if G be found as follows : on P^ take Q so that 



P Q = -PoP lt and on OP* take G so that QG = ^ QP 2 . 

 n n+ i 



(17) Show that Ex. (15) is a special case of Ex. (16). 



(18) Prove the following rule for constructing the geometric sum of 

 n vectors P (} P 1} PoPz, PoPs, PoP n issuing from the same point P : 

 on P^ take G l so that P^ \P^P Z \ on G& take G 2 so that 

 G 1 G 2 = $G 1 P 3 ; on G^ take G 3 so that G 2 G 3 = G 2 P 4 ; and so on. 

 If G be the last point so determined, the geometric sum of the n vectors 

 is =nP G. 



