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 of the following gives possible values of - and -

The correct option is A α =1, β=1 a⃗,b⃗ and c⃗ are coplanar. #160; = [a⃗b⃗c⃗]=0 =α +β=2 .... (1) Also a⃗ bisects the ang ...

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Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

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 of the following gives possible values of $α$ and $β$

α = 2, β = 1

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α = 1, β = 1

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α = 2, β = 2

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α = 1, β = 2

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Solution

The correct option is A $α = 1, β = 1$
$\to a, \to b$ and $\to c$ are coplanar.
$= [\to a \to b \to c] = 0$
$= α + β = 2$ .... $(1)$
Also $\to a$ bisects the angle between $\to b$ and $\to c$ .
$\Rightarrow \to a = λ (^b +^c)$

$= \to a = λ (\frac{^i + 2^j +^k}{\sqrt{2}})$ ... $(2)$

Comparing $(2)$ with $\to a = α^i + 2^j + β^k$ , we get
$λ = \sqrt{2}$ , $α = 1$ and $β = 1$ , which also satisfies $(1)$

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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 of the following gives possible values of α and β

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