Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.
Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

Find the vector equation of the line passing through the point with position vector and parallel to the vector . Deduce the Cartesian equations of the line.
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:

$\stackrel{^}{r}=2\stackrel{^}{i}+\stackrel{^}{ı}–5\stackrel{^}{k}+\lambda \left(\stackrel{^}{i}+–3\stackrel{^}{i}–\stackrel{^}{k}\right)$
\hat{r}=2 \hat{i}+\hat{\imath}-5 \hat{k}+\lambda(\hat{i}+-3 \hat{i}-\hat{k})

Cartesian form:

$\frac{x–2}{1}=\frac{y–1}{3}=\frac{z+5}{–1}$
\frac{x-2}{1}=\frac{y-1}{3}=\frac{z+5}{-1}