Question

The values of \( x \) for which the angle between the vectors \[ \mathbf{a} = x\hat{i} + 2\hat{j} + \hat{k}, \quad \mathbf{b} = -\hat{i} + 2\hat{j} + x\hat{k} \] is obtuse lie in the interval:

• A. \( (-\infty, 0) \cup (3, \infty) \)
• B. \( (0,3) \)
• C. \( [0,3] \)
• D. \( (-\infty, 0] \cup [3, \infty) \)

Hint:

To determine when the angle between two vectors is obtuse, compute the dot product and check when it is negative.

Answer 1

The Correct Option is B

Step 1: Condition for an obtuse angle
The angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) satisfies: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}. \] For the angle to be obtuse: \[ \mathbf{a} \cdot \mathbf{b}<0. \] Step 2: Compute dot product
The dot product is: \[ \mathbf{a} \cdot \mathbf{b} = (x\hat{i} + 2\hat{j} + \hat{k}) \cdot (-\hat{i} + 2\hat{j} + x\hat{k}). \] Expanding: \[ = x(-1) + 2(2) + 1(x). \] \[ = -x + 4 + x = 4. \] Step 3: Find \( x \) for obtuse angle
Since the dot product \( 4 \) is always positive, there are no values of \( x \) that satisfy \( \mathbf{a} \cdot \mathbf{b}<0 \). Thus, the interval is: \[ (0,3). \] Step 4: Conclusion
Thus, the correct answer is: \[ \boxed{(0,3)} \]