Question
Find the vector equation of the line passing through the point having position vector
$-\hat{i}-\hat{j}+2 \hat{k}$ and parallel to the line $\bar{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}+\hat{k})$.
$-\hat{i}-\hat{j}+2 \hat{k}$ and parallel to the line $\bar{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}+\hat{k})$.
The required line is parallel to the line
$\bar{r}=(\hat{i}+2 \hat{j}+3 \hat{k}+2(3 \hat{i}+2 \hat{j}+\hat{k})$
$\therefore$ it is parallel to the vector
$\bar{b}=3 \hat{i}+2 \hat{j}+\hat{k}$
The vector equation of the line passing through $A(\bar{a})$ and parallel to $\bar{b}$ is $\bar{r}=\bar{a}+\lambda \bar{b}$ where
λ is a scalar. ∴ the required vector equation of the line is
$\overline{\mathrm{r}}=(-\hat{i}-\hat{j}+2 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}+\hat{k})$
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