Question

If the scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with the unit vector in the direction of the resultant of the vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $\lambda \hat{i}+2\hat{j}+3\hat{k}$, is unity, then $\lambda$ equals

• A. 5
• B. 2
• C. 1
• D. -1

Answer 1

The correct option is C 1

$\overrightarrow{r}=(2\hat{i}+4\hat{j}-5\hat{k})+(\lambda \hat{i}+2\hat{j}+3\hat{k})$

$=(2+\lambda )\hat{i}+6\hat{j}-2\hat{k}$

$\overrightarrow{e}$ = unit vector in the direction of $\overrightarrow{r}$

$=\frac{(2+\lambda )\hat{i}+6\hat{j}-2\hat{k}}{\sqrt{(2+\lambda {)}^2+36+4}}$

Therefore

$\overrightarrow{e}.(\hat{i}+\hat{j}+\hat{k})=1$

$\Rightarrow (2+\lambda )+6-2=\sqrt{(2+\lambda {)}^2+40}$

$\Rightarrow (\lambda +6{)}^2=(2+\lambda {)}^2+40$

$\Rightarrow 12\lambda +36=4\lambda +44$

$\Rightarrow 8\lambda =8$

$\Rightarrow \lambda =1$