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Q.
The cartesian equation of the line $l$ when it passes through the point $(x_1, y_1, z_1)$ and parallel to the vector $\vec{b}=a\hat{i}+b\hat{j}+c\hat{k}$, is
Three Dimensional Geometry
Solution:
The coordinates of the given point $A$ be $(x_1, y_1, z_1)$ Consider the coordinates of any point $P$ on the line be $(x, y, z)$.
The line is parallel to $\vec{b}=a\hat{i}+b\hat{j}+c\hat{k}$
Hence, the direction ratio of the line are $a$, $b$ and $c$.
$\therefore $ Cartesian eq. of a line through the point $\left(x_{1}, y_{1} z_{1}\right)$
and having direction ratios $a$, $b$ and $c$ is
$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$