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

The correct option is A α=1,β=1 Given the vectors, a⃗ =αî +2ĵ +βk̂ b⃗ =î +ĵ c⃗ =ĵ +k̂ Since, the vectors lie in one plane. #160; #160; ...

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Standard IX

Mathematics

Applications of Dot Product

The vector ...

Question

The vector $\to a = α^i + 2^j + β^k$ lies in the plane of the vectors $\to b =^i +^j$ and $\to c =^j +^k$ and bisects the angle between $\to b$ and $\to c$ . Then , which one if 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 A $α = 1, β = 1$
Given the vectors,
$\to a = α^i + 2^j + β^k \to b =^i +^j \to c =^j +^k$
Since, the vectors lie in one plane.
$∴ ∣ ∣ ∣ ∣ \begin{matrix} α & 2 & β 1 & 1 & 0 0 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow α (1) - 2 (1) + β (1) = 0 \Rightarrow α + β = 2$
For, $α + β = 2, α m u s t b e 1 a n d β m u s t b e 1$
$∴$ the possible values of $α$ and $β$ are $1$ and $1$ respectively.

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The vector →a=α^i+2^j+β^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 if the following gives possible values of α and β?

The correct option is A α=1,β=1Given the vectors,→a=α^i+2^j+β^k→b=^i+^j→c=^j+^kSince, the vectors lie in one plane. ∴∣∣ ∣∣α2β110011∣∣ ∣∣=0⇒α(1)−2(1)+β(1)=0⇒α+β=2For, α+β=2,αmustbe1andβmustbe1∴ the possible values of α and β are 1 and 1 respectively.

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