If the vectors -k- -k- and -k- lie on a plane- where - - - and - are distinct non-negative numbers- then - is

The correct option is C Geometric mean of α and β Since, the vectors are coplanar. ∴[ α amp;α amp;γ; 1 amp;0 amp;1; γ amp;γ amp;β ]= ...

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Standard XII

Mathematics

Standard Deviation about Mean for Frequency Distributions

If the vector...

Question

If the vectors $α^i + α^j + γ^k,^i +^k$ and $γ^i + γ^j + β^k$ lie on a plane, where $α, β$ and $γ$ are distinct non-negative numbers, then $γ$ is

Arithmetic mean of

α

and

β

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Geometric mean of

α

and

β

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Harmonic mean of

α

and

β

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None of the above

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Solution

The correct option is C Geometric mean of $α$ and $β$
Since, the vectors are coplanar.
$∴ ⎡ ⎢ ⎣ \begin{matrix} α & α & γ 1 & 0 & 1 γ & γ & β \end{matrix} ⎤ ⎥ ⎦ = 0$
or, $α (0 - γ) + α (γ - β) + γ (γ - 0) = 0$
or, $- α γ + α γ - α β + γ^{2} = 0$
or, $γ^{2} = α β$
or, $γ = \sqrt{α β}$
Therefore, $γ$ is GM of $α$ and $β .$
Hence, B is the correct option.

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Similar questions

Q. Let

α, β

γ

be distinct real numbers. The points whose position vectors are

α^i + β^j + γ^k

β^i + γ^j + α^k

and

γ^i + α^j + β^k

Q. The vector

\to a = α^i + 2^j + β^k

lies in the plane of the vectors

\to b =^i +^j

and

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and bisects the angle between

\to b

and

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Q. 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

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and

β

Q. If

\frac{tan (α - β)}{tan α} + \frac{{sin}^{2} γ}{{sin}^{2} α} = 1,

then prove that tan

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β

.
i.e., than

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Q. If

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and

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α \neq β \neq γ

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If the vectors α^i+α^j+γ^k,^i+^k and γ^i+γ^j+β^k lie on a plane, where α,β and γ are distinct non-negative numbers, then γ is

The correct option is C Geometric mean of α and βSince, the vectors are coplanar.∴⎡⎢⎣ααγ101γγβ⎤⎥⎦=0or, α(0−γ)+α(γ−β)+γ(γ−0)=0or, −αγ+αγ−αβ+γ2=0or, γ2=αβor, γ=√αβTherefore, γ is GM of α and β.Hence, B is the correct option.

If the vectors $α^i + α^j + γ^k,^i +^k$ and $γ^i + γ^j + β^k$ lie on a plane, where $α, β$ and $γ$ are distinct non-negative numbers, then $γ$ is