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Question

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 ___

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Solution

Here, s=4p+3q+5r(i)and s=(p+q+r)x+(pq+r)y+(pq+r)z(ii)4p+3q+5r=p(x+yz)+q(xyz)+r(x+y+z)
On comparing both sides, we get
x+yz=4, xyz=3 and x+y+z=5
On solving above equations, we get
x=4,y=92,z=722x+y+z=8+9272=9


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