If x-c-y-a and y-c-x-b- where c is a non - zero vector- then which of the following is correct

Given: x+c×y=a⋯(i) y+c×x=b⋯(ii) Taking cross with c, of equation (ii) We have c×y+c× (c×x)=c×b Now, using (i), we have ⇒ (a x)+ (c·x)c (c·c)x=c×b ⋯(iii) nbs ...

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Byju's Answer

Standard XII

Physics

Work Done as Dot Product

If x+c×y=a an...

Question

If $\to x + \to c \times \to y = \to a$ and $\to y + \to c \times \to x = \to b$ , where $\to c$ is a non - zero vector, then which of the following is correct

\to x = \frac{(\to b \times \to c) + \to a + (\to c \cdot \to a) \to c}{1 + \to c \cdot \to c}

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\to x = \frac{(\to c \times \to b) + \to b + (\to c \cdot \to a) \to c}{1 + \to c \cdot \to c}

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\to x = \frac{(\to a \times \to c) + \to b + (\to c \cdot \to a) \to c}{1 + \to c \cdot \to c}

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\to x = \frac{(\to a \times \to c) + \to b + (\to c \cdot \to a) \to c}{1 - \to c \cdot \to c}

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Solution

The correct option is A $\to x = \frac{(\to b \times \to c) + \to a + (\to c \cdot \to a) \to c}{1 + \to c \cdot \to c}$
Given:
$\to x + \to c \times \to y = \to a \dots (i) \to y + \to c \times \to x = \to b \dots (i i)$
Taking cross with $\to c,$ of equation $(i i)$ We have $\to c \times \to y + \to c \times (\to c \times \to x) = \to c \times \to b$
Now, using $(i),$ we have
$\Rightarrow (\to a - \to x) + (\to c \cdot \to x) \to c - (\to c \cdot \to c) \to x = \to c \times \to b \dots (i i i)$

Also taking dot product with $\to c$ in equation $(i),$ we have
$\Rightarrow \to c \cdot \to x + \to c \cdot (\to c \times \to y) = \to c \cdot \to a$ or $\to c \cdot \to x + 0 = \to c \cdot \to a$ or $\to c \cdot \to x = \to c \cdot \to a \dots (i v)$

Using $(i v)$ in $(i i i),$ we have
$\Rightarrow (\to a - \to x) + (\to c \cdot \to a) \to c - (\to c \cdot \to c) \to x = \to c \times \to b$
$\Rightarrow \to x (1 + (\to c \cdot \to c)) = \to b \times \to c + \to a + (\to c \cdot \to a) \cdot \to c$
$\Rightarrow \to x = \frac{(\to b \times \to c) + \to a + (\to c \cdot \to a) \cdot \to c}{1 + \to c \cdot \to c}$

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If →x+→c×→y=→a and →y+→c×→x=→b, where →c is a non - zero vector, then which of the following is correct

If $\to x + \to c \times \to y = \to a$ and $\to y + \to c \times \to x = \to b$ , where $\to c$ is a non - zero vector, then which of the following is correct