The angle between two unit vectors $ extbf{q}$ and $ extbf{b}$ is $ heta$ and $| extbf{a} + extbf{b}| < 1$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2022 - Paper 1

To determine the possible range of the angle $heta$ between the two unit vectors $extbf{q}$ and $extbf{b}$, we start by considering the condition that the magnitude of the sum of the two vectors must be less than 1, i.e., $| extbf{a} + extbf{b}| < 1$. This implies that the vectors must be somewhat opposite in direction.

Because both $extbf{q}$ and $extbf{b}$ are unit vectors, the maximum value of their sum occurs when they are perfectly aligned (i.e., $heta = 0$). In contrast, when they are opposite to each other, the sum reaches a minimum, which has a magnitude of 0.

Given that the combined magnitude is limited to less than 1, this indicates that $heta$ must be in the range of angles that allow for a non-maximum sum.

Hence, the range of possible angles $heta$ that satisfies $| extbf{a} + extbf{b}| < 1$ corresponds to the interval where the angle is less than rac{2 heta}{3} but greater than or equal to 0. Thus, the correct answer is:

B. $0 \, \leq \, \theta \< \frac{2\pi}{3}$