Step 1: Understanding the Concept:
The problem asks for the relationship between the coefficients of unit vectors when they form a vector along the angle bisector of two given vectors.
Step 2: Key Formula or Approach:
For any two non-zero vectors \( \vec{a} \) and \( \vec{b} \), the vector along their internal angle bisector is proportional to the sum of their corresponding unit vectors.
Angle bisector vector \( \vec{v} = \lambda \left( \hat{a} + \hat{b} \right) = \lambda \left( \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} \right) \), for some scalar \( \lambda>0 \).
Step 3: Detailed Explanation:
The given vector along the angle bisector is:
\[ \vec{v} = x \frac{\overline{a}}{|\overline{a}|} + y \frac{\overline{b}}{|\overline{b}|} \]
We know from vector properties that the internal bisector of the angle between vectors \( \overline{a} \) and \( \overline{b} \) is parallel to the vector \( \frac{\overline{a}}{|\overline{a}|} + \frac{\overline{b}}{|\overline{b}|} \).
This means any vector along this bisector must be a scalar multiple of this sum:
\[ \vec{v} = \lambda \left( \frac{\overline{a}}{|\overline{a}|} + \frac{\overline{b}}{|\overline{b}|} \right) = \lambda \frac{\overline{a}}{|\overline{a}|} + \lambda \frac{\overline{b}}{|\overline{b}|} \]
Comparing the coefficients of the given vector with the general form, we have:
\[ x = \lambda \]
\[ y = \lambda \]
Therefore, \( x \) must be equal to \( y \).
This implies \( x = y \), which can be written as \( x - y = 0 \).
Step 4: Final Answer:
The relation between x and y is \( x - y = 0 \).