Find the domain of λ for which the vector r=(λ2−4)i+2j+(λ2−9)k makes acute angles with the co-ordinate axes.
A
λϵ{(−3,−1)∪[0,3]}
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B
λϵ{(−3,0)∪[1,3]}
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C
λϵ{(−3,−2)∪[0,3]}
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D
λϵ{(−3,−2)∪[2,3]}
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Solution
The correct option is Dλϵ{(−3,−2)∪[2,3]} Since →r makes acute angle with ^i,^j,^k we have →r.^i>0,→r.^j>0, and →r.^k>0 ∴λ2−4>0,2>0,λ2−9>0 ∴(λ+2)(λ−2)>0 and (λ+3)(λ−3)>0 ∴−2<λ or λ>2 and −3<λ or λ>3 ∴λϵ(−3,−2) or λϵ(2,3) ∴λϵ{(−3,−2)∪[2,3]}