Question

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

• A. $x-x_1=y-y_1=z-z_1$
• B. $x+x_1=y+y_1=z-z_1$
• C. $\frac{x+x_1}{a}=\frac{y+y_1}{b}=\frac{z+z_1}{c}$
• D. $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Answer 1

Answer: (D)

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
$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$