If $\vec a,\vec b$ are unit vectors and $ heta$ is the angle between them, show that (i) $\sin\dfrac heta2=\dfrac12|\vec a-\vec b|$ (ii) $\cos\dfrac heta2=\dfrac12|\vec a+\vec b|$ (iii) $ an\dfrac heta2=\dfrac{|\vec a-\vec b|}{|\vec a+\vec b|}$.

Question

If $\vec a,\vec b$ are unit vectors and $	heta$ is the angle between them, show that (i) $\sin\dfrac	heta2=\dfrac12|\vec a-\vec b|$ (ii) $\cos\dfrac	heta2=\dfrac12|\vec a+\vec b|$ (iii) $	an\dfrac	heta2=\dfrac{|\vec a-\vec b|}{|\vec a+\vec b|}$.

Accepted Answer

**Step 2.** Using $1-\cos	heta=2\sin^2\!\left(\dfrac	heta2ight)$: $$|\vec a-\vec b|^2=4\sin^2\!\left(\frac	heta2ight)\ \Rightarrow\ |\vec a-\vec b|=2\sin\!\left(\frac	heta2ight)\ \Rightarrow\ \sin\frac	heta2=\frac12|\vec a-\vec b|.\quad	ext{(i)}$$

**Step 3.** Similarly, $|\vec a+\vec b|^2=1+1+2\cos	heta=2(1+\cos	heta)=4\cos^2\!\left(\dfrac	heta2ight)$ (using $1+\cos	heta=2\cos^2(	heta/2)$), so $$|\vec a+\vec b|=2\cos\!\left(\frac	heta2ight)\ \Rightarrow\ \cos\frac	heta2=\frac12|\vec a+\vec b|.\quad	ext{(ii)}$$ …

Q.If $\vec a,\vec b$ are unit vectors and $\theta$ is the angle between them, show that

Concept understanding — Scalar (Dot) Product

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Q.If $\vec a,\vec b$ are unit vectors and $\theta$ is the angle between them, show that

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