Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line

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Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line

Find the Cartesian and vector equations of a line which passes through the point $(1,2,3)$ and is parallel to the line $\frac{-\mathrm{X}-2}{1}=\frac{\mathrm{y}+3}{7}=\frac{2 Z-6}{3}$
Answer
Given: line passes through $(1,2,3)$ and is parallel to the line

\frac{-x-2}{1}=\frac{y+3}{7}=\frac{2 z-6}{3}

To find: equation of line in Vector and Cartesian form
Formula Used: Equation of a line is
Vector form: $\vec{r}=\vec{a}+\lambda \vec{b}$
Cartesian form: $\frac{x-x_{1}}{b_{n}}=\frac{y-y_{1}}{b_{z}}=\frac{z-z_{1}}{k_{x}}=z$ line.
Explanation:
Since the line (say $L_{1}$ ) is parallel to another line (say $\left.L_{2}\right), L_{1}$ has the same direction ratios as that of $\mathrm{L}_{2}$
Here, $\frac{-q}{\lambda}=\hat{i}-2 \hat{i}+\overrightarrow{j k}$
Equation of $L_{2}$ can be rewritten as: