Mathematics

Mathematics

Problem

Let two non-collinear unit vectors a^\hat{a} and b^\hat{b} form an acute angle. A point PP moves so that at any time tt the position vector OP⃗\vec{OP} (where OO is the origin) is given by a^cos⁡t+b^sin⁡t\hat{a}\cos t + \hat{b}\sin t. When PP is farthest from the origin OO, let MM be the length of OP⃗\vec{OP} and u^\hat{u} be the unit vector along OP⃗\vec{OP}. a. u^=a^+b^∣a^+b^∣\hat{u} = \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|} and M=(1+a^⋅b^)1/2M = (1 + \hat{a} \cdot \hat{b})^{1/2} b. u^=a^−b^∣a^−b^∣\hat{u} = \frac{\hat{a} - \hat{b}}{|\hat{a} - \hat{b}|} and M=(1+a^⋅b^)1/2M = (1 + \hat{a} \cdot \hat{b})^{1/2} c. u^=a^+b^∣a^+b^∣\hat{u} = \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|} and M=(1+2a^⋅b^)1/2M = (1 + 2\hat{a} \cdot \hat{b})^{1/2} d. u^=a^−b^∣a^−b^∣\hat{u} = \frac{\hat{a} - \hat{b}}{|\hat{a} - \hat{b}|} and M=(1+2a^⋅b^)1/2M = (1 + 2\hat{a} \cdot \hat{b})^{1/2}

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