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

The correct option is B λ μ =1 a1 b1 c are unitvector a #160; b #160; = 0 (given) ⇒ (a b) #160; · (b + c) = 0 ⇒ a ...

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Byju's Answer

Standard XII

Mathematics

Fundamental Theorem In 3D

If a⃗, b⃗, ...

Question

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

μ^{2} + ω^{2} = 1

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λ - μ = 1

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(μ + 1)^{2} + μ^{2} + ω^{2} = 1

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λ^{2} + μ^{2} = 1

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Solution

The correct option is B $λ - μ = 1$
$- \to a 1 \to b 1 c a r e u n i t v e c t o r$
$\to a - \to b = 0 (g i v e n)$
$\Rightarrow (\to a \to - \to b) \cdot (b + c) = 0$
$\Rightarrow \to a \cdot - \to a + \to a \cdot b \to - \to c \cdot \to b - {∣ ∣ \to c ∣ ∣}^{2} = 0$
$\Rightarrow \to a \cdot \to b + a \cdot c - \to b \cdot \to c = 1$
$\Rightarrow 0 + \to a \cdot \to c - \to b \cdot \to c = 1$
$\Rightarrow \to a \cdot ¯ ¯ c - \to b \cdot \to c = 1 e q n . (i)$
Then
$\to \subset = λ \to a + μ \to b + ω (\to a \times ¯ ¯ b)$
$\Rightarrow T a k e s c a l a r p r o d u c t w i t h a a n d b$
$\Rightarrow \to a \cdot \to c = λ + 0 + 0$
$\Rightarrow \to a \cdot \to c = λ e q n . (i i)$
$[∵ \to a \cdot (\to a \times \to b) = 0]$
$[∵ ∣ ∣ \to a ∣ ∣ = ∣ ∣ ∣ \to b ∣ ∣ ∣ = ∣ ∣ \to c ∣ ∣ = 1 u n i t e c t o r]$
$\Rightarrow \to b \cdot \to c = λ (\to a - \to b) + μ {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} + ω \to b \cdot (\to a \times \to b)$
$\Rightarrow \to b \cdot \to c = 0 + μ + 0$
$\Rightarrow \to b \cdot \to c = μ e q n . (i i i)$
$o n su b s t r a c t i n g e q^{η} e q n (i i) f r o m (i i i)$
$\Rightarrow \to a \cdot ¯ ¯ c - \to b \cdot \to c = λ - μ$
$f r o m e q n (i)$
$\Rightarrow 1 = λ - μ$
$∴ 1 + μ = λ$

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Similar questions

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

Q. 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

Nonzero vectors $\to a, \to b, \to c$ satisfy $\to a . \to b$ =0, $(\to b - \to a) . (\to b + \to c)$ = 0 and 2 $| \to b + \to c | = | \to b - \to a |$ .