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Condition for Two Lines to be on the Same Plane

Let c=xî+yĵ+z...

Question

Let $\to c = x^i + y^j + z^k$ is the external angle bisector between two vectors $\to a = 2^i -^j + 3^k$ and $\to b =^i + 2^j - 3^k, | \to c | = 3 \sqrt{46} .$ Then $x + y + z =$

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Solution

Given vectors, $\to a = 2^i -^j + 3^k$ and $\to b =^i + 2^j - 3^k,$
external angle bisector is $λ (^a -^b)$
$\to c = λ (\frac{2^i -^j + 3^k}{\sqrt{14}} - \frac{^i + 2^j - 3^k}{\sqrt{14}}) = \frac{λ}{\sqrt{14}} (^i - 3^j + 6^k) \Rightarrow | \to c | = \frac{\sqrt{46} λ}{\sqrt{14}} = 3 \sqrt{46} \Rightarrow λ = 3 \sqrt{14} \Rightarrow \to c = 3^i - 9^j + 18^k ∴ x + y + z = 3 - 9 + 18 = 12$

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Similar questions

\to a = 2^i +^j +^k, \to b = 3^i - 4^j + 2^k, \to c =^i - 2^j + 2^k

then the projection of

\to a + \to b

\to c

\to a = 3^i +^j - 2^k; \to b =^i + λ^j - 3^k

\to a ⊥ \to b

, find value of

λ

(x, y, z) \neq (0, 0, 0)

(^i +^j + 3^k) x + (3^i - 3^j +^k) y + (- 4^i + 5^j) z = λ (x^i + y^j + z^k)

, then the value of

λ

Q. The values of

λ

(x, y, z) \neq (0, 0, 0)

(^i +^j + 3^k) x + (3^i - 3^j +^k) y + (- 4^i + 5^j) z = λ (x^i + y^j + z^k)

\to c = x^i + y^j + z^k

is the internal angle bisector between

\to a = 2^i -^j +^k

\to b =^i + 2^j -^k

| \to c | = \sqrt{40} .

Then the value of

(x + y + z) =

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Condition for Two Lines to be on the Same Plane

Standard XII Mathematics

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