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.

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:

\hat{r}=2 \hat{i}+\hat{\imath}-5 \hat{k}+\lambda(\hat{i}+-3 \hat{i}-\hat{k})

Cartesian form:

$\frac{x\u20132}{1}=\frac{y\u20131}{3}=\frac{z+5}{\u20131}$\frac{x-2}{1}=\frac{y-1}{3}=\frac{z+5}{-1}