Question

$\text{If}\overrightarrow{a}=\hat{i}+\lambda \hat{j}+2\hat{k};\overrightarrow{b}=\mu \hat{i}+\hat{j}-\hat{k}$ are orthogonal and $|\overrightarrow{a}|=|\overrightarrow{b}|$, then $(\lambda ,\mu )=$

• A. $(\frac{1}{4},\frac{7}{4})$
• B. $(\frac{7}{4},\frac{1}{4})$
• C. $(\frac{1}{4},\frac{9}{4})$
• D. $(\frac{-1}{4},\frac{9}{4})$

Answer 1

The correct option is A $(\frac{1}{4},\frac{7}{4})$

$\overrightarrow{a}$ and $\overrightarrow{b}$ are orthogonal.
$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=0$
$\Rightarrow \mu +\lambda -2=0\cdots \left(i\right)$
$\text{Also},|\overrightarrow{a}|=|\overrightarrow{b}|$
$\Rightarrow 1+{\lambda }^2+4={\mu }^2+1+1$
$\Rightarrow {\lambda }^2-{\mu }^2+3=0\cdots \left(ii\right)$
$\text{From}\left(i\right)\text{and}\left(ii\right)$,
$\lambda =\frac{1}{4},\mu =\frac{7}{4}$