Question

# If the vectors $$\vec{a}=i-j+2k; \vec{b}=2i+4j+k$$ and $$\vec{c}=2013\lambda i+j-\mu k$$ are mutually orthogonal then $$(\lambda, \mu)$$=...

A
(32013;2)
B
(32013;2)
C
(32013;2)
D
(32013;2)

Solution

## The correct option is A $$(\frac{-3}{2013};-2)$$$$\vec{a}=i-\hat{j}+2k$$$$\vec{b}=2i-4j+k$$$$\vec{c}=2013 \lambda+j-\mu k$$$$\therefore\vec{a}$$ & $$\vec{c}$$ orthogonal $$\Rightarrow \vec{a} .\vec{c}=0$$$$\Rightarrow 2013\lambda -1-2\mu =0………1$$Also , $$\vec{b}$$ & $$\vec{c}$$ are orthogonal$$\Rightarrow \vec{b} , \vec{c} =0$$$$\Rightarrow 4026\lambda +4-\mu =0……2$$Now solving these equation, we getequation 1 $$\times 2$$ equation 2$$4026\lambda -2-4\mu =0\\ 4026\lambda +4-\mu =0\\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad \quad \quad -6-3\mu =0\\ \quad \quad \quad \quad \quad \quad 3u=-6 \Rightarrow\boxed { \mu =-2 }$$Also ,$$2013\lambda -1+4=0$$$$\lambda=\dfrac{-3}{2013}= \dfrac{-1}{671} \quad \therefore \left(\dfrac{-3}{2013},-2\right)$$$$\Rightarrow$$ option $$\left(A\right)$$ is correctMaths

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