EXAMPLES 23 



6. Three vectors of lengths P, P, and P V2 meet in a point and are mutu- 

 ally at right angles. Determine the magnitude of the resultant and the angles 

 between its direction and that of each component. 



7. Three vectors of lengths P, 2 P, 3 P meet in a point and are directed 

 along the diagonals of the three faces of a cube meeting at the point. Determine 

 the magnitude of their resultant. 



8. Show that the resultant of three vectors represented by the diagonals of 

 three faces of a parallelepiped meeting in a vertex A is represented by twice 

 the diagonal of the parallelepiped drawn from A. 



9 D is a point in the plane of the triangle ABC, and I is the center of its 

 inscribed circle. Show that the resultant of the vectors a AD, b BD, c CD is 

 (a + b + c) ID, where a, 6, c are the lengths of the sides of the triangle. 



10. ABCD, A'B'C'Dr are two parallelograms in the same plane. Find the 

 resultant of vectors drawn from a point proportional to and in the same direc- 

 tion as AA', B'B, (7(7, D'D. 



11. If is the center of the circumscribed circle of the triangle ABC and 

 P its orthocenter, show that OP is the resultant of the vectors OA, OB, and 

 OC ; also that 2 PO is the resultant of PA, PB, PC. 



12. The chords AOB and COD of a circle intersect at right angles. Show 

 that the resultant of the vectors OA, OB, OC, OD is represented by twice the 

 vector OP, where P is the center of the circle. 



GENERAL EXAMPLES 



(In these examples take the acceleration produced by gravity to be 32 feet per second 



per second) 



1. A point possesses simultaneously velocities of 2,3,8 feet per second, 

 in the directions of a point describing the three sides of an equilateral tri- 

 angle in order. Find the magnitude of its velocity. 



2. A point possesses simultaneously velocities, each equal to v, in the 

 directions of lines drawn from the center of a regular hexagon to five of 

 its angular points. Find the magnitude and direction of the resultant 

 velocity. 



3. When a steamer is in motion it is found that an awning 8 feet above 

 the deck protects from rain the portion of the deck more than 4 feet behind 

 the vertical projection of the edge of the awning ; but when the steamer 

 comes to rest the line of separation of the wet and dry parts is 6 feet in 

 front of this projection. Find the velocity of the steamer, if that of the 

 rain be 20 feet per second. 



4. A ship sailing along the equator from east to west finds that from 

 noon one day (local time) to noon the next day (local time) the distance 

 covered is 420 miles. What would be the day's run, if the ship were sail- 

 ing at the same rate from west to east ? 



