Question
Let two non-collinear unit vectors \( \hat{a} \) and \( \hat{b} \) form
\( \mathrm{P} \)
an acute angle. A point \( P \) moves so that at any time the position vector \( \overline{O P} \) (where \( O \) is the origin) is given by \( \hat{a} \cos t+\hat{b} \sin t \). When \( P \) is farthest from origin \( O \), let \( M \) be the length of \( \overline{O P} \) and \( \hat{u} \) be the unit vector along \( \overline{O P} \). Then,
(1) \( \hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|} \) and \( M=(1+\hat{a} \cdot \hat{b})^{1 / 2} \)
(2) \( \hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|} \) and \( M=(1+\hat{a} \cdot \hat{b})^{1 / 2} \)
(3) \( \hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|} \) and \( M=(1+2 \hat{a} \cdot \hat{b})^{1 / 2} \)
(4) \( \hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|} \) and \( M=(1+2 \hat{a} \cdot \hat{b})^{1 / 2} \)
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