Question

The cartesian equation of the plane whose vector equation is
$\gamma =(1+\lambda -\mu )\hat{i}+(2-\lambda )\hat{j}+(3-2\lambda +2\mu )\hat{k}$
where $\lambda ,\mu$ are scalars, is

• A. 2 x+y=5
• B. 2 x-y=5
• C. 2 x-z=5
• D. 2 x+z=5

Answer 1

Answer: (D)

Given,
$\gamma =(1+\lambda -\mu )\hat{i}+(2-\lambda )\hat{j}+(3-2\lambda +2\mu )\hat{k}$
$=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}-2\hat{k})+\mu (-\hat{i}+2\hat{k})$
Equation of plane is given by
$\{r-(\hat{i}+2\hat{j}+3\hat{k})\}\cdot [(\hat{i}-\hat{j}-2\hat{k})\times (-\hat{i}+2\hat{k})]=0$
$\Rightarrow [r-(\hat{i}+2\hat{j}+3\hat{k})]\cdot (-2\hat{i}-\hat{k})=0$
$\Rightarrow r\cdot (2\hat{i}+\hat{k})-5=0$
$\Rightarrow r\cdot (2\hat{i}+\hat{k})=5$
Now, equation of plane in cartesian form is
$(x\hat{i}+y\hat{j}+z\hat{k})\cdot (2\hat{i}+\hat{k})=5$
$\therefore 2x+z=5$