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: Express the line in parametric form.
The given line equation is:\[\frac{x - 2}{2} = \frac{y - \frac{5}{2}}{-\frac{3}{2}} = z + 1.\]This indicates the line passes through the point:\[\left(2, \frac{5}{2}, -1\right),\]with direction ratios:\[(2, -\frac{3}{2}, 0).\]Step 2: Determine 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:\[\vec{b} = 2\hat{i} - \frac{3}{2}\hat{j} + p\hat{k}.\]As the \( z \)-component of the direction ratios is \( 0 \), we set:\[p = 0.\] Final Answer:\[\boxed{p = 0}\]