Question

If the vector equation of the line \[ \frac{x - 2}{2} = \frac{2y - 5}{-3} = z + 1, \] is given by: \[ \vec{r} = \left(2\hat{i} + \frac{5}{2}\hat{j} - \hat{k}\right) + \lambda\left(2\hat{i} - \frac{3}{2}\hat{j} + p\hat{k}\right), \] then \( p \) is equal to:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

When working with vector equations, compare the position vector and direction ratios to find any unknown components.

Answer 1

The Correct Option is A

Step 1: Write the parametric form of the line.
The given line equation can be expressed as: \[ \frac{x - 2}{2} = \frac{y - \frac{5}{2}}{-\frac{3}{2}} = z + 1. \]
From this, we can determine that the line passes through the point: \[ \left(2, \frac{5}{2}, -1\right), \] and has direction ratios: \[ (2, -\frac{3}{2}, 0). \] Step 2: Match the vector equation.
The position vector of the point is: \[ \vec{a} = 2\hat{i} + \frac{5}{2}\hat{j} - \hat{k}. \]
The direction vector is: \[ \vec{b} = 2\hat{i} - \frac{3}{2}\hat{j} + p\hat{k}. \] Since the \( z \)-component of the direction ratios is \( 0 \), we equate: \[ p = 0. \] Final Answer: \[ \boxed{p = 0} \]