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 $$\overrightarrow{OP}$$(where O is the origin) is given by $$\hat{a} \cos t + \hat{b} \sin t$$. When is farthest from origin O, let M be the length of $$\overrightarrow{OP}$$ and $$\hat{u}$$ be the unit vector along $$\overrightarrow{OP}$$ . Then

$$\hat{u} = \frac{\hat{a} + \hat{b}}{\mid \hat{a} + \hat{b} \mid}$$ and $$M = (1 + \hat{a}.\hat{b})^{\frac{1}{2}}$$

$$\hat{u} = \frac{\hat{a} - \hat{b}}{\mid \hat{a} - \hat{b} \mid}$$ and $$M = (1 + \hat{a}.\hat{b})^{\frac{1}{2}}$$

$$\hat{u} = \frac{\hat{a} + \hat{b}}{\mid \hat{a} + \hat{b} \mid}$$ and $$M = (1 + 2\hat{a}.\hat{b})^{\frac{1}{2}}$$

$$\hat{u} = \frac{\hat{a} - \hat{b}}{\mid \hat{a} - \hat{b} \mid}$$ and $$M = (1 + 2\hat{a}.\hat{b})^{\frac{1}{2}}$$