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Answer the following.
If a⃗ and b⃗ are unit vectors enclosing an angle θ and |a⃗+b⃗| < 1, then find the values between…
If $\vec{a}$ and $\vec{b}$ are unit vectors enclosing an angle $\theta$ and $|\vec{a}+\vec{b}| < 1$, then find the values between which $\theta$ lies. (Understanding)
Answer
Answer
Official answer key$|\vec a + \vec b|^2 < 1 \Rightarrow |\vec a|^2 + |\vec b|^2 + 2\vec a\cdot\vec b < 1 \Rightarrow 2 + 2\cos\theta < 1 \Rightarrow \cos\theta < -\frac{1}{2}$. Since $-1 \le \cos\theta < -\frac12$ and $0 \le \theta \le \pi$, $\frac{2\pi}{3} < \theta \le \pi$.
Ans: $\theta \in \left(\frac{2\pi}{3}, \pi\right]$