Question

If \(\overset{̄}{a}\) and \(\overset{̄}{b}\) are unit vectors and \(θ\) (\(0 < θ < π\)) is the angle between them, then the value of \(\frac{|\overset{̄}{a}+\overset{̄}{b}|}{|\overset{̄}{a}-\overset{̄}{b}|}\) is equal to...

• A. \(tan\frac{θ}{2}\)
• B. \(sin\frac{θ}{2}\)
• C. \(cos\frac{θ}{2}\)
• D. \(cot\frac{θ}{2}\)

Hint:

Square both modulus values and use 1 + cos t = 2cos^2(t/2).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
For any vector, \(|\vec v|^2=\vec v\cdot\vec v\). For unit vectors, \(\vec a\cdot\vec a=\vec b\cdot\vec b=1\) and \(\vec a\cdot\vec b=\cos\theta\).

Step 2: Square of the sum:
\[ |\vec a+\vec b|^2=1+1+2\cos\theta=2(1+\cos\theta)=4\cos^2\tfrac\theta2 \]

Step 3: Square of the difference:
\[ |\vec a-\vec b|^2=2-2\cos\theta=4\sin^2\tfrac\theta2 \]

Step 4: Ratio:
For \(0<\theta<\pi\), both \(\cos\frac\theta2\) and \(\sin\frac\theta2\) are positive. So
\[ \frac{|\vec a+\vec b|}{|\vec a-\vec b|}=\frac{2\cos\frac\theta2}{2\sin\frac\theta2}=\cot\frac\theta2 \]

Step 5: Choose:
Option (D). The tangent in option (A) is the reciprocal.

Final Answer:
The ratio is cot(theta/2). \[ \boxed{\cot\frac\theta2} \]