Let \( \mathrm{OAB} \) be a regular triangle with side length unity ( \( O \) being the origin)....

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Let \( \mathrm{OAB} \) be a regular triangle with side length unity ( \( O \) being the origin). Also \( M, N \) are the points of trisection of \( A B, M \) being closer to \( A \) and \( N \) closer to \( B \). Position vectors of \( A, B, M \) and \( N \) are
\( \mathrm{P} \) \( \vec{a}, \vec{b}, \vec{m} \) and \( \vec{n} \) respectively. Which of the following hold(s) good?

W
(A) \( \overrightarrow{\mathrm{m}}=\mathrm{x} \overrightarrow{\mathrm{a}}+\mathrm{y} \overrightarrow{\mathrm{b}} \Rightarrow \frac{2}{3} \) and \( y=\frac{1}{3} \)
(B) \( \overrightarrow{\mathrm{m}}=\mathrm{xa}+\mathrm{y} \overrightarrow{\mathrm{b}} \Rightarrow \mathrm{x}=\frac{5}{6} \) and \( \mathrm{y}=\frac{1}{6} \)
(C) \( \overrightarrow{\mathrm{m}} . \overrightarrow{\mathrm{n}} \) equals \( \frac{13}{18} \)
(D) \( \overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}} \) equals \( \frac{15}{18} \)
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