Suppose that `vec p,vecqand vecr` are three non- coplaner in `R^(3)` ,Let the components of a vector`vecs` along `vecp , vec q and vecr` be 4,3, and 5, respectively , if the components this vector `vec s` along `(-vecp+vec q +vecr),(vecp-vecq+vecr) and (-vecp-vecq+vecr)` are x, y and z , respectively , then the value of `2x+y+z` is

Options

A. 7
B. 8
C. 9
D. 6

Answer

C

Explanation

It is given that the components of vectors `vecs " along " vecp, vec q and vec r ` are 4, 3 and 5 respectively.
`therefore vecs = 4vec p + 3 vecq + 5 vec r " " ` …(i)
The components of `vec s ` along `-vec p + vec q + vec r , vec p - vec q + vec r and - vec p - vec q + vec r` and x, y and z respectively.
` therefore vec s = x (- vec p + vec q + vec r ) +y( vec p - vec q + vec r ) + z ( - vec p - vec q + vec r ) `
`rArr vec s = (-x+y-z) vec p + (x-y-z) vec q + (x+y+z) vec r " " ` ...(ii)
From (i) and (ii) , we obtain
`-x+y-z=4, x-y-z=3 and x+y+z=5`
Solving these equations, we obtain
`x=4, y=(9)/(2), z=(7)/(2)`
`therefore 2x+y+z=8+(9)/(2)-(7)/(2)=9.`