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{\imath}+\hat{\jmath}-3 \hat{k}+\lambda(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})

Cartesian form:

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