Question

Direction ratios of the straight line \( \vec{r} = 2\hat{i} - 3\hat{j} + \hat{k} + m(9\hat{i} - 2\hat{j} + 5\hat{k}) \) are:

• A. \( <2, -3, 1> \)
• B. \( <9, 2, 5> \)
• C. \( <-2, 3, -1> \)
• D. \( <9, -2, 5> \)

Hint:

The vector multiplied by the parameter (like \( m \), \( \lambda \), or \( \mu \)) always represents the direction of the line.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The vector equation of a line is \( \vec{r} = \vec{a} + \lambda \vec{b} \), where \( \vec{a} \) is the position vector of a point on the line and \( \vec{b} \) is the direction vector of the line.
Step 2: Detailed Explanation:
The direction ratios of the line are the scalar components of the direction vector \( \vec{b} \).
In the given equation:
\[ \vec{r} = (2\hat{i} - 3\hat{j} + \hat{k}) + m(9\hat{i} - 2\hat{j} + 5\hat{k}) \] Here, the direction vector \( \vec{b} = 9\hat{i} - 2\hat{j} + 5\hat{k} \).
The components are \( 9, -2, 5 \).
Thus, the direction ratios are \( <9, -2, 5> \).
Step 3: Final Answer:
The direction ratios are \( <9, -2, 5> \).