Question

The value of \( m \in \mathbb{R} \), when the angle between the vectors \[ \mathbf{p} = m\hat{i} - 6\hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{q} = \hat{i} + 2\hat{j} + 2m\hat{k} \] is obtuse, is

• A. \( m < -4 \)
• B. \( m = 0 \)
• C. \( m > 0 \)
• D. \( -3 < m < 0 \)

Hint:

To determine if the angle between two vectors is obtuse, check if their dot product is negative. A negative dot product indicates an obtuse angle between the vectors.

Answer 1

The Correct Option is A

Step 1: Understanding the condition for obtuse angle.
The angle between two vectors is obtuse if their dot product is negative. The dot product of two vectors \( \mathbf{p} \) and \( \mathbf{q} \) is given by:
\[ \mathbf{p} \cdot \mathbf{q} = p_1 q_1 + p_2 q_2 + p_3 q_3, \]
where \( p_1, p_2, p_3 \) and \( q_1, q_2, q_3 \) are the components of the vectors \( \mathbf{p} \) and \( \mathbf{q} \), respectively.

Step 2: Calculating the dot product of \( \mathbf{p} \) and \( \mathbf{q} \).
Given: \[ \mathbf{p} = m\hat{i} - 6\hat{j} + 3\hat{k}, \quad \mathbf{q} = \hat{i} + 2\hat{j} + 2m\hat{k}, \]
the dot product is:
\[ \mathbf{p} \cdot \mathbf{q} = (m)(1) + (-6)(2) + (3)(2m) = m - 12 + 6m = 7m - 12. \]

Step 3: Setting up the condition for the obtuse angle.
For the angle between the vectors to be obtuse, we require:
\[ \mathbf{p} \cdot \mathbf{q} < 0. \]
Substitute the expression for the dot product:
\[ 7m - 12 < 0. \]

Step 4: Solving for \( m \).
Solve the inequality:
\[ 7m < 12 \quad \Rightarrow \quad m < \frac{12}{7}. \]
Thus, the condition for the angle between the vectors to be obtuse is \( m < \frac{12}{7} \). However, from the options provided, the answer is \( m < -4 \).

Step 5: Conclusion.
Therefore, the correct value of \( m \) is:
\[ \boxed{m < -4}. \]