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Question

Suppose that p,q and r are three non-coplanar vectors in R3 . Let the components of vector 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) , (pq+r) and (pq+r) are x,y and z respectively , then the value of 2x+y+z is

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Solution

Given s=4p+3q+5r -----(1)

As components of s along (p+q+r),(pq+r),(pq+r) are x,y,z respectively.

s=x(p+q+r)+y(pq+r)+z(pq+r)

s=p(x+yz)+q(xyz)+r(x+y+z) ----(2)

Comparing (1) and (2) gives

x+yz=4,xyz=3,x+y+z=5

solving above three equations gives x=4,y=92,z=72

2x+y+z=8+9/27/2=9

2x+y+z=9



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