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Condition for Coplanarity of Four Points

If c=xî+yĵ+zk...

Question

If $\to c = x^i + y^j + z^k$ is the internal angle bisector between $\to a = 2^i -^j +^k$ and $\to b =^i + 2^j -^k$ and $| \to c | = \sqrt{40} .$ Then the value of $(x + y + z) =$

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Solution

Given vectors : $\to a = 2^i -^j +^k$ and $\to b =^i + 2^j -^k$
internal angle bisector is given by,
$\to c = λ (^a +^b) = λ (\frac{2^i -^j +^k}{\sqrt{6}} + \frac{^i + 2^j -^k}{\sqrt{6}}) \Rightarrow \to c = λ \cdot \frac{3^i +^j}{\sqrt{6}}$
Since, $| \to c | = \sqrt{40}$
$\Rightarrow \frac{10 λ^{2}}{6} = 40 \Rightarrow λ = 2 \sqrt{6} ∴ \to c = 6^i + 2^j \Rightarrow (x + y + z) = 8$

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Condition for Coplanarity of Four Points

Standard XII Mathematics

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