A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.
A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

A line passes through the point and is parallel to the vector . Find the equations of the line in vector and Cartesian forms.
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:

$\stackrel{\to }{\mathrm{r}}=2\stackrel{^}{ı}+\stackrel{^}{ȷ}–3\stackrel{^}{k}+\lambda \left(\stackrel{^}{ı}–2\stackrel{^}{ȷ}+3\stackrel{^}{k}\right)$
\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}