If a- b- c- are unit vectors such that a-b-0 and a-c- -b-c-0 and c-a-b-a-b- for some scalars - - -then

The correct options are C λ^2+μ^2+ω^2=1 D λ=μ+1a.b=0 ⇒a×b is a unit vector. c.a=λ, c.b=μ c.(a×b)=ω (a c).(b+c)=0 ⇒λ μ 1=0 λ=μ+1 |c|^2=λ^2|a|^2+μ^2 |b|^2+w^ ...

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Byju's Answer

Standard XII

Mathematics

Condition for Coplanarity of Four Points

If a⃗, b⃗, c⃗...

Question

If $\to a, \to b, \to c$ are unit vectors such that $\to a . \to b = 0$ and $(\to a - \to c) . (\to b + \to c) = 0$ and $\to c = λ \to a + μ \to b + ω (\to a \times \to b)$ for some scalars $λ, μ, ω$ , then

μ = λ + 1

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λ^{2} + μ^{2} + ω^{2} = 2

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λ^{2} + μ^{2} + ω^{2} = 1

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λ = μ + 1

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Solution

The correct options are
C $λ^{2} + μ^{2} + ω^{2} = 1$
D $λ = μ + 1$
$\to a . \to b = 0 \Rightarrow \to a \times \to b$ is a unit vector.

$\to c . \to a = λ$ , $\to c . \to b = μ$

$\to c . (\to a \times \to b) = ω$

$(\to a - \to c) . (\to b + \to c) = 0 \Rightarrow λ - μ - 1 = 0$

$λ = μ + 1$

$| \to c |^{2} = λ^{2} | \to a |^{2} + μ^{2} | \to b |^{2} + w^{2} | \to a \times \to b |^{2}$

$(\to a, \to b, \to a \times \to b are orthogonal pairs of unit vectors)$

$1 = λ^{2} + μ^{2} + w^{2}$

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