The vector -2-k- lies in the plane of vectors b- and c-k- and bisects the angle between b- and c- Which one of the following gives possible values of - and -

The correct option is D α=1, β=1 b⃗.(αî+2ĵ+ βk̂)=|b̂|(√(α ^2+4)+β^2)cosθ (α +2)=(√(2))(√(α ^2+4+ β^2))cosθ ————(1) a⃗.(αî)+αĵ+ βk̂=|a⃗|(√(2^ ...

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Byju's Answer

Standard XII

Physics

Unit Vectors

The vector ...

Question

The vector $α^i + 2^j + β^k$ lies in the plane of vectors $\to b =^i +^j$ and $\to c =^j +^k$ and bisects the angle between $\to b$ and $\to c$ Which one of the following gives possible values of $α$ and $β$

α = 2, β = 2

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α = 1, β = 2

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α = 2, β = 1

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α = 1, β = 1

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Solution

The correct option is D $α = 1, β = 1$
$\to b . (α^i + 2^j + β^k) = |^b | (\sqrt{α^{2} + 4} + β^{2}) cos θ$
$(α + 2) = (\sqrt{2}) (\sqrt{α^{2} + 4 + β^{2}}) cos θ$ ------------(1)
$\to a . (α^i) + α^j + β^k = | \to a | (\sqrt{2^{2} + 4 + β^{2}}) cos θ$
$(α + β) = (\sqrt{2}) \sqrt{α^{2} + 4 + β^{2}} cos θ$ --------------(2)
$\frac{α + 2}{2 + β} = 1$
$α = β$ ------------(3)
$2^i + 2^j + β^k = μ (^i +^j) + λ (^j +^k)$ (collineorlity)
$μ + λ = 2$
$μ = α λ = β$
$α + β = 2$ --------------(4)
$α = 1 β = 1$

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The vector α^i+2^j+β^k lies in the plane of vectors →b=^i+^j and →c=^j+^k and bisects the angle between →b and →c Which one of the following gives possible values of α and β

The vector $α^i + 2^j + β^k$ lies in the plane of vectors $\to b =^i +^j$ and $\to c =^j +^k$ and bisects the angle between $\to b$ and $\to c$ Which one of the following gives possible values of $α$ and $β$