Suppose that p , q and r are three non-coplanar vectors in R ^3 . Let the components of a vector s along p , q and r be 4,3 and 5 respectively. If the components of this vector s along (- p + q + r ),( p - q + r ) and (- p - q + r ) are x , y and z respectively, then the value of 2 x+y+z is

Step-by-step solution

Given aligned & s =4 p +3 q +5 r & ~s =(- p + q + r ) x+( p - q + r ) y+(- p - q + r ) z & aligned & 4 p +3 q +5 r =(-x+y- z ) p +(x-y- z ) q &+(x+y+ z ) r aligned aligned Comparing, we get aligned & -x+y-z=4 ...(i) & x-y-z=3 ...(ii) & x+y+z=5 ...(iii) aligned Solving (i), (ii) and (iii), we get x=4, y= 9 2 , z= -7 2 aligned 2 x+y+z & =2(4)+ 9 2 - 7 2 & =2(4)+1 & =9 aligned