Question

If the line \[ \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) \quad \text{is parallel to the plane} \quad \vec{r} \cdot (3i - 2j + mk) = 10, \] then the value of \( m \) is

• A. 2
• B. -3
• C. -2
• D. 3

Hint:

For parallelism between a line and a plane, the direction vector of the line must be perpendicular to the normal vector of the plane.

Answer 1

The Correct Option is C

Step 1: Understand the condition for parallelism.
For the line \( \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) \) to be parallel to the plane \( \vec{r} \cdot (3i - 2j + mk) = 10 \), the direction vector of the line must be perpendicular to the normal vector of the plane. The direction vector of the line is \( \vec{d} = (2, 1, 2) \), and the normal vector of the plane is \( \vec{n} = (3, -2, m) \).
Step 2: Apply the perpendicularity condition.
For perpendicular vectors, their dot product must be zero: \[ \vec{d} \cdot \vec{n} = 2 \cdot 3 + 1 \cdot (-2) + 2 \cdot m = 0. \] This simplifies to: \[ 6 - 2 + 2m = 0 \quad \Rightarrow \quad 4 + 2m = 0 \quad \Rightarrow \quad m = -2. \]
Step 3: Conclusion.
Thus, the value of \( m \) is -2, corresponding to option (C).