Let a vector make an obtuse angle with the vector and an angle , with the positive z-axis. If the set of all possible values of , then is equal to _

Question

Let a vector $\vec{a} = \sqrt{2} \hat{i} - \hat{j} + λ \hat{k}, λ > 0,$ make an obtuse angle with the vector $\vec{b} = - λ^{2} \hat{i} + 4 \sqrt{2} \hat{j} + 4 \sqrt{2} \hat{k}$ and an angle $θ, \frac{π}{6} < θ < \frac{π}{2}$ , with the positive z-axis. If the set of all possible values of $λ is (α, β) - {γ}$ , then $α + β + γ$ is equal to _______. [2026]

Smart Achievers · Accepted Answer

(5)

$\frac{\vec{a} \cdot \hat{k}}{| \vec{a} |} = \cos θ = \frac{λ}{\sqrt{3 + λ^{2}}} = \cos θ$

$\Rightarrow 0 < \frac{λ}{\sqrt{3 + λ^{2}}} < \frac{\sqrt{3}}{2}$

$\Rightarrow λ > 0 & 4 λ^{2} < 9 + 3 λ^{2} \Rightarrow λ^{2} < 9$

$\Rightarrow λ \in (0, 3) . . . (1)$

$\Rightarrow \vec{a} \cdot \vec{b} < 0$ $\Rightarrow - \sqrt{2} λ^{2} - 4 \sqrt{2} + 4 \sqrt{2} λ < 0$

$\Rightarrow λ^{2} - 4 λ + 4 > 0$ $\Rightarrow {(λ - 2)}^{2} > 0$

$\Rightarrow λ \neq 2 . . . (2)$

From (1) and (2)

$λ \in (0, 3) - {2}$

$∴$ $α = 0, β = 3, γ = 2$

$\Rightarrow α + β + γ = 5$

Solution

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Solution

Q. Let a vector a→=2i^-j^+λk^, λ>0, make an obtuse angle with the vector b→=-λ2i^+42j^+42k^ and an angle θ, π6<θ<π2, with the positive z-axis. If the set of all possible values of λ is (α,β)-{γ}, then α+β+γ is equal to _______. [2026]

Ans. (5) a→·k^|a→|=cosθ=λ3+λ2=cosθ ⇒0<λ3+λ2<32 ⇒λ>0 & 4λ2<9+3λ2⇒λ2<9 ⇒λ∈(0,3) ...(1) ⇒a→·b→<0⇒-2λ2-42+42λ<0 ⇒λ2-4λ+4>0⇒(λ-2)2>0 ⇒λ≠2 ...(2) From (1) and (2) λ∈(0,3)-{2} ∴ α=0, β=3, γ=2 ⇒α+β+γ=5

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