28 GEOMETRY OF MOTION. [56 



sums be ^X, S F, ^Z. The vectors are therefore equivalent to 

 the three vectors ^X y ^Y y ^Z, which form the concurrent edges 

 of a rectangular parallelepiped whose diagonal drawn through 

 the origin O is the resultant vector OR = R, i.e. 



R = 

 The direction of this vector is given by the equations 



, cos J3=2, cos 7 =H 



where a, /?, .7 are the angles made by OR with the axes 6V, Oy> 

 Oz, respectively. 



If all the vectors lie in the same plane, we have simply : 



2 , tan a = 



56. Exercises. 



(1) A ship sails first 5 miles N. 30 E., then 12 miles N. 60 E., and 

 finally 25 miles E. 75 S. Find distance and bearing of the point 

 reached : (a) graphically, (b) analytically. 



(2) Is a scale of 8 miles to the inch sufficient to obtain the results of 

 Ex. ( i) correctly to whole miles and degrees ? 



(3) A rigid body undergoes three translations, of i, 2, and 3 feet,, 

 whose directions are respectively parallel to the three sides of an equi- 

 lateral triangle taken the same way round. Find the . resulting dis- 

 placement. 



(4) A ship is carried by the current 2 miles due W., and at the same 

 time by the wind 4 miles due N.E., and by her screw n miles E. 30 

 S. Find her resultant displacement. 



(5) A ferry-boat crosses a river in a direction inclined at an angle of 

 60 to the direction of the current. If the width of the river be half a 

 mile, what are the component displacements of the boat along the river 

 and at right angles to it ? 



(6) Two vectors of equal length a are inclined to each other at an 

 angle a. Find the resultant in magnitude and direction. 



(7) For what angle a, in Ex. (6), is the resultant equal in magni- 

 tude : (a) to each component a ? (b) to J a ? 



