Question

Equation of a plane parallel to the vectors $2\hat{i}+\hat{j}+\hat{k},\hat{i}+2\hat{j}+3\hat{k}$ and passing through the point $\hat{i}+2\hat{j}+\hat{k}$ is

• A. $(x-1)\hat{i}+(y-2)\hat{j}+(z-1)\hat{k}=$$\lambda (2\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}+3\hat{k})$
• B. $(x-2)\hat{i}+(y-2)\hat{j}+(z-1)\hat{k}=$$\lambda (\hat{i}+2\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}+3\hat{k})$
• C. $(x+1)\hat{i}+(y+2)\hat{j}+(z+1)\hat{k}=$$\lambda (2\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}+\hat{k})$
• D. $(x-1)\hat{i}+(y+2)\hat{j}+(\hat{z}+1)\hat{k}=\lambda$$(2\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}+\hat{k})$

Answer 1

The correct option is A $(x-1)\hat{i}+(y-2)\hat{j}+(z-1)\hat{k}=$$\lambda (2\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}+3\hat{k})$

Parallel vectors are $2\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$
Equation passing through the point $\hat{i}+2\hat{j}+\hat{k}$
So equation of plane parallel to the vectors and passing through the point is $\hat{r}=\hat{i}+2\hat{j}+\hat{k}+\mu (2\hat{i}+\hat{j}+\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k})$
$(x-1)\hat{i}+(y-2)\hat{j}+(z-1)\hat{k}=(2\mu +\lambda )\hat{i}+(\mu +2\lambda )\hat{j}+(\mu +3\lambda )\hat{k}$