Question

Find the vector equation of the line which passes through the point $(3,4,5)$ and is parallel to the vector $2\hat{i}+2\hat{j}-3\hat{k}.$

Answer 1

The vector equation of a line which passes through the point $\overrightarrow{a}$ and parallel to the vector $\overrightarrow{b}$ is given as
$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b},$ where $\lambda$ is a scalar.
$\text{Here,}\overrightarrow{a}=3\hat{i}+4\hat{j}+5\hat{k}\text{and}\overrightarrow{b}=2\hat{i}+2\hat{j}-3\hat{k}$
So, Required equation is $\overrightarrow{r}=3\hat{i}+4\hat{j}+5\hat{k}+\lambda (2\hat{i}+2\hat{j}-3\hat{k})$