The vector a - -i - 2j - -k lies in the plane of the vectors b - i - j and c - j - k and bisects the angle between b and c - Then- which one of the following gives possible values of - and - -

The correct option is D α = 1, β = 1 a⃗ =λ( b⃗ +c⃗ #160; ) (∵a⃗ lies in plane of b⃗ and c⃗ and bisects the angle between b ...

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Byju's Answer

Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

The vector ...

Question

The vector $\to a = α ˆ i + 2 ˆ j + β ˆ k$ lies in the plane of the vectors $\to b = ˆ i + ˆ j$ and $\to c = ˆ j + ˆ k$ and bisects the angle between $\to b$ and $\to c .$ Then, 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 $α = 1, β = 1$
$\to a = λ (\to b + \to c)$
$(∵ \to a$ lies in plane of $\to b$ and $\to c$ and bisects the angle between $\to b$ and $\to c)$
$\Rightarrow α \to i + 2 \to j + β \to k = λ (\frac{\to i + \to j}{\sqrt{2}} + \frac{\to j + \to k}{\sqrt{2}})$
$\Rightarrow α \to i + 2 \to j + β \to k = λ (\frac{\to i + 2 \to j + \to k}{\sqrt{2}})$
$\Rightarrow λ = \sqrt{2} α, λ = \sqrt{2} a n d λ = \sqrt{2} β$
$∴ α = β = 1$

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The vector →a=αˆi+2ˆj+βˆk lies in the plane of the vectors →b=ˆi+ˆj and →c=ˆj+ˆk and bisects the angle between →b and →c. Then, which one of the following gives possible values of α and β?

The correct option is D α=1,β=1→a=λ(→b+→c)(∵→a lies in plane of →b and →c and bisects the angle between →b and →c)⇒α→i+2→j+β→k=λ(→i+→j√2+→j+→k√2)⇒α→i+2→j+β→k=λ(→i+2→j+→k√2)⇒λ=√2α,λ=√2andλ=√2β∴α=β=1

The vector $\to a = α ˆ i + 2 ˆ j + β ˆ k$ lies in the plane of the vectors $\to b = ˆ i + ˆ j$ and $\to c = ˆ j + ˆ k$ and bisects the angle between $\to b$ and $\to c .$ Then, which one of the following gives possible values of $α$ and $β ?$