Let two non-collinear vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point P moves, so that at any time t the position vector $\overline{OP}$, where O is origin, is given by $\hat{a}\sin t+\hat{b}\cos t.$ when P is farthest from origin O, let M be the length of OP and $\hat{u}$ be the unit vector along $\overline{OP,}$ then
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Logic Tip: The sum of two vectors scaled identically (like multiplying both by $1/\sqrt{2}$) will point in the exact same direction as the unscaled sum. Hence, the unit vector of $k\vec{A}$ is simply the unit vector of $\vec{A}$ (for any $k>0$).