A line passes through the point and is parallel to the vector . Find the equations of the line in vector and Cartesian forms.
Answer
Given: line passes through and is parallel to
To find: equation of line in vector and Cartesian forms
Formula Used: Equation of a line is
Vector form:
Cartesian form:
where is a point on the line and is a vector parallel to the line.
Explanation:
Here, and
Therefore,
Vector form:

$\overrightarrow{\mathrm{r}}=2\hat{ı}+\hat{ȷ}–3\hat{k}+\lambda (\hat{ı}–2\hat{ȷ}+3\hat{k})$
\overrightarrow{\mathrm{r}}=2 \hat{\imath}+\hat{\jmath}-3 \hat{k}+\lambda(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})

Cartesian form:

$\frac{\mathrm{x}–2}{1}=\frac{\mathrm{y}–1}{–2}=\frac{\mathrm{z}+3}{3}$
\frac{\mathrm{x}-2}{1}=\frac{\mathrm{y}-1}{-2}=\frac{\mathrm{z}+3}{3}