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Let two non-collinear unit vectors \( \hat{\mathbf{a}} \) and \( \hat{\mathbf{b}} \) form an acute angle. A point \( P \) moves, so that at any time \( t \)
\( \mathrm{P} \) the position vector \( \overrightarrow{\mathrm{OP}} \) (where, \( O \) is the origin) is

W given by \( \hat{\mathbf{a}} \cos t+\hat{\mathbf{b}} \sin t \). When \( P \) is farthest from origin \( O \), let \( M \) be the length of \( \overrightarrow{\mathbf{O P}} \) and \( \hat{\mathbf{u}} \) be the unit vector along \( \overrightarrow{\mathbf{O P}} \). Then,
(2008, 3M)
(a) \( \hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|} \) and \( M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2} \)
(b) \( \hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\hat{\mathbf{a}}-\hat{\mathbf{b}}|} \) and \( M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2} \)
(c) \( \hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|} \) and \( M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2} \)
(d) \( \hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\hat{\mathbf{a}}-\hat{\mathbf{b}}|} \) and \( M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2} \)
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