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)

$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$

Answer 2

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 e of a line through the point $\left(x_{1}, y_{1} z_{1}\right)$ and having direction ratios $a$, $b$ and $c$ is $\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$