1. Let's understand where the number 0.707 comes from. 2. The value 0.707 is approximately equal to $\frac{1}{\sqrt{2}}$. 3. This number often appears in trigonometry, especially with angles of 45 degrees or $\frac{\pi}{4}$ radians. 4. For example, $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \approx 0.707$. 5. This happens because in a right triangle with equal legs, the hypotenuse is $\sqrt{2}$ times the length of each leg. 6. So, the ratio of one leg to the hypotenuse is $\frac{1}{\sqrt{2}}$, which is about 0.707. 7. This is why 0.707 appears frequently in problems involving 45-degree angles or normalized vectors.