Question

The symmetric form of the equation of the straight line \[ \overrightarrow{r} = \hat{i} + t \hat{j}, \quad t \in \mathbb{R}, \] is:

• A. \( \frac{x-1}{0} = \frac{y}{1} = \frac{z}{0} \)
• B. \( \frac{x}{1} = \frac{y}{1} = \frac{z-1}{0} \)
• C. \( \frac{x-1}{0} = \frac{y-1}{0} = \frac{z}{1} \)
• D. \( \frac{x-1}{1} = \frac{y}{1} = \frac{z}{0} \)
• E. \( \frac{x-1}{0} = \frac{y}{1} = \frac{z}{1} \)

Hint:

In the symmetric form of a line equation, the point on the line is represented as \( (x_1, y_1, z_1) \), and the direction ratios are used in the denominators. If the direction vector has zero components, it will appear as a zero in the corresponding term.

Answer 1

The Correct Option is A

The given equation of the line is in vector form: \[ \overrightarrow{r} = \hat{i} + t \hat{j}, \quad t \in \mathbb{R}. \] This represents a line passing through the point \( (1, 0, 0) \) with direction vector \( (0, 1, 0) \). 
The symmetric form of a line equation is given by: \[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}, \] where \( (x_1, y_1, z_1) \) is a point on the line and \( (a, b, c) \) is the direction vector. 
For this case, the point on the line is \( (1, 0, 0) \), and the direction vector is \( (0, 1, 0) \). Thus, the symmetric form of the equation is: \[ \frac{x - 1}{0} = \frac{y}{1} = \frac{z}{0}. \] 
Thus, the correct answer is option (A).