Question

If the line \( \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) \) is parallel to the plane \( \vec{r} = (3i - 2j - mk) = 5 \), then the value of \( m \) is

• A. \( -2 \)
• B. \( -3 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For parallelism between a line and a plane, use the condition that the dot product of the direction ratios of the line and the normal vector of the plane must be zero.

Answer 1

The Correct Option is C

Step 1: Find the direction ratio of the line.
The direction ratios of the line are given by \( 2, 1, 2 \).
Step 2: Find the normal vector to the plane.
The normal vector to the plane is \( (3, -2, -m) \).
Step 3: Use the parallelism condition.
For parallelism, the direction ratios of the line and the normal vector of the plane must be perpendicular. Hence, their dot product must be zero.
\[ (2)(3) + (1)(-2) + (2)(-m) = 0 \] \[ 6 - 2 - 2m = 0 \] Step 4: Solve for \( m \).
\[ 4 - 2m = 0 \Rightarrow m = 2 \] Step 5: Conclusion.
The value of \( m \) is \( 2 \).