Question

Find the vector equation of a line passing through (2, −1, 1) and parallel to the line whose equations are $\frac{x-3}{2}=\frac{y+1}{7}=\frac{z-2}{-3}.$

Answer 1

We know that the vector equation of a line passing through a point with position vector $\overrightarrow{a}$ and parallel to the vector $\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$.

Here,
$$\begin{gathered} \overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k} \\ \overrightarrow{b}=2\hat{i}+7\hat{j}-3\hat{k} \end{gathered}$$

Vector equation of the required line is
$$\begin{gathered} \overrightarrow{r}=\left(2\hat{i}-\hat{j}+\hat{k}\right)+\lambda \left(2\hat{i}+7\hat{j}-3\hat{k}\right) \\ \mathrm{Here},\lambda \mathrm{is}\mathrm{a}\mathrm{parameter}. \end{gathered}$$