If vector x satisfying x-a-x-bc-d is given by x-a-a-a-d-ca-c-a-2- then the value of -

Given: nbsp; x×a+(x·b)c=d ⇒{x×a+(x·b)c}×c=d×c ⇒ (x×a)×c+(x·b)(c×c)=d×c ⇒ nbsp; (x·c)a (a·c)x=d×c ⇒ nbsp;a×{(x·c)a (a·c)x}=a×{d×c} ⇒ (a·c)(a×x)=a×{d×c} ...

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Byju's Answer

Standard XII

Mathematics

Intersection of Sets

If vector x s...

Question

If vector $\to x$ satisfying $\to x \times \to a + (\to x \cdot \to b) \to c = \to d$ is given by $\to x = λ \to a + \to a \times \frac{\to a \times (\to d \times \to c)}{(\to a \cdot \to c) | \to a |^{2}},$ then the value of $λ =$

\frac{(\to a \cdot \to b)}{| \to a |^{2}}

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\frac{(\to a \cdot \to x)}{| \to a |^{2}}

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\frac{(\to a \cdot \to c)}{| \to a |^{2}}

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\frac{(\to a \cdot \to d)}{| \to a |^{2}}

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Solution

The correct option is B $\frac{(\to a \cdot \to x)}{| \to a |^{2}}$
Given: $\to x \times \to a + (\to x \cdot \to b) \to c = \to d$
$\Rightarrow {\to x \times \to a + (\to x \cdot \to b) \to c} \times \to c = \to d \times \to c \Rightarrow (\to x \times \to a) \times \to c + (\to x \cdot \to b) (\to c \times \to c) = \to d \times \to c \Rightarrow (\to x \cdot \to c) \to a - (\to a \cdot \to c) \to x = \to d \times \to c \Rightarrow \to a \times {(\to x \cdot \to c) \to a - (\to a \cdot \to c) \to x} = \to a \times {\to d \times \to c} \Rightarrow - (\to a \cdot \to c) (\to a \times \to x) = \to a \times {\to d \times \to c} \Rightarrow \to x \times \to a = \frac{\to a \times (\to d \times \to c)}{\to a \cdot \to c} \Rightarrow \to a \times (\to x \times \to a) = \to a \times \frac{\to a \times (\to d \times \to c)}{\to a \cdot \to c} \Rightarrow (\to a \cdot \to a) \to x - (\to a \cdot \to x) \to a = \to a \times \frac{\to a \times (\to d \times \to c)}{\to a \cdot \to c} \Rightarrow \to x = \frac{(\to a \cdot \to x) \to a}{| \to a |^{2}} + \to a \times \frac{\to a \times (\to d \times \to c)}{(\to a \cdot \to c) | \to a |^{2}} \dots (i)$
Comparing $(i)$ with $\to x = λ \to a + \to a \times \frac{\to a \times (\to d \times \to c)}{(\to a \cdot \to c) | \to a |^{2}},$
$∴ λ = \frac{(\to a \cdot \to x)}{| \to a |^{2}}$

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