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.

Answer

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}

\Rightarrow-4: 1:-1 \\

\Rightarrow 4:-1: 1

\end{array}

So,

Therefore,

Vector form:

\overrightarrow{\mathrm{r}}=3 \hat{\mathbf{i}}-2 \hat{\imath}+\hat{k}+\lambda(4 \hat{\imath}-\hat{\jmath}+\hat{k})

Cartesian form:

$\frac{x\u20133}{4}=\frac{y+2}{\u20131}=\frac{z\u20131}{1}$\frac{x-3}{4}=\frac{y+2}{-1}=\frac{z-1}{1}