EXAMPLES V 77 



EXAMPLES V 



(1) The co-ordinates of a point P are ( - 8, 5, 7). If O 

 is the origin, find the length of OP and the angles OP makes 

 with the axes of reference. 



(2) The co-ordinates of a point P are (2-1, 8*4, 4*7). If O is 

 the origin, find the length of OP and the angles OP makes with 

 the axes of reference. 



(8) The co-ordinates of a point P are ( - 8'2, 5-1, 3*9) . If O 

 is the origin, find the length of OP and the angles OP makes with 

 the axes of reference. 



(4) The co-ordinates of a point P are (2, 4, 6), and of a point 

 Q (3, - 7, 5). Find the length of PQ and the angles PQ makes 

 with the axes of reference. 



(5) The co-ordinates of a point P are (1-8, 5*3, 2-9), and of a 

 point Q (3-7, 2-9, 5-4). Find the length of PQ and the angles 

 PQ makes with the axes of icference. 



(6) P and Q are two points whose co-ordinates are (3, 7, 2) and 

 (5, 3, 7) respectively. Find the length of the perpendicular 

 drawn to the line PQ from the origin. 



(7) The co-ordinates of three points P, Q, and R are (3, 2, 4), 

 (5, 4, 7), and (4, 7, 2) respectively. Find the lengths of the sides, 

 the area, and the angles of the triangle PQR. 



(8) The co-ordinates of three points P, Q, R are (5, 2, 4), 

 (7, 5, 2), and (9, 3, 7) respectively. Find the angle between the 

 lines PQ and QR, and also the perpendicular distance from the 

 point R to the line PQ. 



(9) For each of the three planes 



x-2y +2-2 = 

 I2x + I5y - 102 - 60 = 

 and 6x + 4y + 3z - 12 = 



Find (a) the intercepts on the axes of reference ; (&) the length 

 of OP, the perpendicular drawn from the origin to the plane ; 

 (c) the angles which OP makes with the axes of reference ; (d) 

 the angles which the plane makes with the axes of reference ; 

 (e) the co-ordinates of P. 



(10) Find the angle between the two planes 



and 4# + 5y + lOz 20 = 



(11) Find the direction cosines of the perpendiculars drawn from 

 the origin to each of the planes 



8x - 2y + 62 - 6 = 

 6x+8y-2z-6 = 



Do these two planes intersect at right angles ? 



