Question

# Let the coordinates of a points p with respect to the system of non-coplanar vectors →a,→b and →c is (3,2,1). Then coordinates of p with respect to the system of vectors →a+→b+→c, →a−→b+→c and →a+→b−→c is

A
(32,12,1)
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B
(32,1,12)
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C
(12,32,1)
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D
(1,32,12)
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Solution

## The correct option is A (32,12,1)coordinate of p with respect to →a,→b and →c=(3,2,1)∴(3,2,1)=3→a+2→b+→c ___ (1)Now, α(→a+→b+→c)+β(→a−→b+→c)+γ(→a+→b+→c)=α→a+α→b+α→c+β→a−β→b+β→c+γ→a+γ→b−→c=(α+β+γ)→a+(α−β+γ)→b+(α+β−γ)→c __ (ii)comparing (i) and (ii), we getα+β+γ=3 ___ (iii)α−β+γ=2 ___ (iv)α+β−γ=1 ___ (v)(iii)−(iv)⇒2β=1⇒β=12(iv)+(v)⇒2α=3α=32(iii)+(iv)⇒2α+2γ=52(32)+2γ=53+2γ=5⇒2γ=5−3⇒2γ=2⇒γ=1∴ coordinate of p with respect to →a+→b+→c,→a−→b+→c and →a+→b−→c=(α,β,γ)=(32,12,1)

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