Find the vector and Cartesian equations of the line passing through the points and , 3).
Answer
Given: line passes through the 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

$\Rightarrow 4:–7:–3$
\Rightarrow 4:-7:-3
$\begin{array}{l}\Rightarrow –4:7:3\\ \text{ So, }\overrightarrow{\mathrm{b}}=–4\hat{i}+7\hat{\mid }+\overrightarrow{3}\hat{k}\end{array}$
\begin{array}{l}
\Rightarrow-4: 7: 3 \\
\text { So, } \overrightarrow{\mathrm{b}}=-4 \hat{i}+7 \hat{\mid}+\overrightarrow{3} \hat{k}
\end{array}

Therefore,
Vector form:

$\overrightarrow{\mathrm{r}}=2\hat{ı}–3\hat{ȷ}+\lambda (–4\hat{ı}+7\hat{ȷ}+3\hat{k})$
\overrightarrow{\mathrm{r}}=2 \hat{\imath}-3 \hat{\jmath}+\lambda(-4 \hat{\imath}+7 \hat{\jmath}+3 \hat{k})

Cartesian form:

$\frac{x–2}{–x}=\frac{y+3}{7}=\frac{z}{3}$
\frac{x-2}{-x}=\frac{y+3}{7}=\frac{z}{3}