Question

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

• A. $(\frac{-3}{2013};-2)$
• B. $(\frac{3}{2013};-2)$
• C. $(\frac{3}{2013};2)$
• D. $(\frac{-3}{2013};2)$

Answer 1

The correct option is A $(\frac{-3}{2013};-2)$

$\overrightarrow{a}=i-\hat{j}+2k$

$\overrightarrow{b}=2i-4j+k$

$\overrightarrow{c}=2013\lambda +j-\mu k$

$\therefore \overrightarrow{a}$ & $\overrightarrow{c}$ orthogonal

$\Rightarrow \overrightarrow{a}.\overrightarrow{c}=0$

$\Rightarrow 2013\lambda -1-2\mu =0\dots \dots \dots 1$

Also , $\overrightarrow{b}$ & $\overrightarrow{c}$ are orthogonal

$\Rightarrow \overrightarrow{b},\overrightarrow{c}=0$

$\Rightarrow 4026\lambda +4-\mu =0\dots \dots 2$

Now solving these equation, we get

equation 1 $\times 2$ equation 2

$4026\lambda -2-4\mu =04026\lambda +4-\mu =0\_\_\_\_\_\_\_\_\_\_\_\_-6-3\mu =03u=-6\Rightarrow \mu =-2$

Also ,$2013\lambda -1+4=0$

$\lambda =\frac{-3}{2013}=\frac{-1}{671}\therefore (\frac{-3}{2013},-2)$

$\Rightarrow$ option $\left(A\right)$ is correct