Suppose that p,q and r are three non-coplanar vectors in R3. 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 2x+y+z is
Here, s=4p+3q+5r⋯(i)and s=(−p+q+r)x+(p−q+r)y+(−p−q+r)z⋯(ii)∴4p+3q+5r=p(−x+y−z)+q(x−y−z)+r(x+y+z)
On comparing both sides, we get
−x+y−z=4, x−y−z=3 and x+y+z=5
On solving above equations, we get
x=4,y=92,z=−72∴2x+y+z=8+92−72=9