The equation of a line passing through the point $(2,-1,1)$ and parallel to the line joining the points…

The equation of a line passing through the point $(2,-1,1)$ and parallel to the line joining the points $\hat{i}+2 \hat{j}+2 \hat{k}$ and $-\hat{i}+4 \hat{j}+\hat{k}$ is
  1. $\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(-2 \hat{i}+2 \hat{j}-\hat{k})$
  2. $\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+6 \hat{j}+3 \hat{k})$
  3. $\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}-2 \hat{j}-\hat{k})$
  4. $\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}-6 \hat{j}-3 \hat{k})$

Solution

Let $\bar{b}=\hat{i}+2 \hat{j}+2 \hat{k}, \quad \bar{c}=-\hat{i}+4 \hat{j}+\hat{k}$ $\Rightarrow \overline{\mathrm{c}}-\overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$
The equation of the line passing through $2 \hat{i}-\hat{j}+\hat{k}$ and parallel to $-2 \hat{i}+2 \hat{j}-\hat{k}$ is $\overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$

Asked in: MHT CET 2024 (15 May Shift 2)

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