Chapter 14: Problem 7
Let two non-collinear unit vectors \(\hat{a}\) and \(\hat{b}\) form an acute angle. A point \(P\) moves so that at any time \(t\) the position vector \(\overrightarrow{O P}\) (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 \(O P\) and \(\hat{u}\) be the unit vector along \(\overrightarrow{O P}\). Then, (A) \(\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\) and \(M=(1+\hat{a} \cdot \hat{b})^{\frac{1}{2}}\) (B) \(\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\) and \(M=(1+\hat{a} \cdot \hat{b})^{\frac{1}{2}}\) (C) \(\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\) and \(M=(1+2 \hat{a} \cdot \hat{b})^{\frac{1}{2}}\) (D) \(\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\) and \(M=(1+2 \hat{a} \cdot \hat{b})^{\frac{1}{2}}\)
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