Question

Let \( \overline{\text{OA}} = \overline{\text{a}} \), \( \overline{\text{OB}} = \overline{\text{b}} \) and if the vector along the angle bisector of \( \angle \text{AOB} \) is given by \( x \frac{\overline{\text{a}}}{|\overline{\text{a}}|} + y \frac{\overline{\text{b}}}{|\overline{\text{b}}|} \) then

• A. \( x - y = 0 \)
• B. \( x + y = 0 \)
• C. \( x = 2y \)
• D. \( y = 2x \)

Hint:

The diagonal of a rhombus (formed by unit vectors) bisects its angles.

Answer 1

The Correct Option is A

Step 1: Concept The angle bisector of two vectors is found by adding their respective unit vectors.

Step 2: Meaning Let \(\hat{a} = \frac{\overline{a}}{|\overline{a}|}\) and \(\hat{b} = \frac{\overline{b}}{|\overline{b}|}\). The vector \(\hat{a} + \hat{b}\) bisects the angle between \(\overline{a}\) and \(\overline{b}\).

Step 3: Analysis The given bisector vector is \(x\hat{a} + y\hat{b}\). For this to be a bisector, the components along the unit vectors must be equal to maintain symmetry.

Step 4: Conclusion Thus, \(x\) must equal \(y\), leading to \(x - y = 0\). Final Answer: (A)