Find the vector equation of a line passing through the point A(3, -2, 1) and parallel to the line joining the points B(-2, 4, 2) and C(2, 3, 3). Also, find the Cartesian equations of the line.
Find the vector equation of a line passing through the point A(3, -2, 1) and parallel to the line joining the points B(-2, 4, 2) and C(2, 3, 3). Also, find the Cartesian equations of the line.

Find the vector equation of a line passing through the point and parallel to the line joining the points and Also, find the Cartesian equations of the line.
Given: line passes through the point and is parallel to the line joining points and
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 with being the direction ratios of the line.
Explanation:
Here,
The direction ratios of the line are

$\begin{array}{l}⇒–4:1:–1\\ ⇒4:–1:1\end{array}$
\begin{array}{l}
\Rightarrow-4: 1:-1 \\
\Rightarrow 4:-1: 1
\end{array}

So,
Therefore,
Vector form:

$\stackrel{\to }{\mathrm{r}}=3\stackrel{^}{\mathbf{i}}–2\stackrel{^}{ı}+\stackrel{^}{k}+\lambda \left(4\stackrel{^}{ı}–\stackrel{^}{ȷ}+\stackrel{^}{k}\right)$
\overrightarrow{\mathrm{r}}=3 \hat{\mathbf{i}}-2 \hat{\imath}+\hat{k}+\lambda(4 \hat{\imath}-\hat{\jmath}+\hat{k})

Cartesian form:

$\frac{x–3}{4}=\frac{y+2}{–1}=\frac{z–1}{1}$
\frac{x-3}{4}=\frac{y+2}{-1}=\frac{z-1}{1}