Vector Algebra
Magnitude Ratio
Grade 12
Question:
<p><strong>Ex. 55</strong> Let A, B, C, D, E represent vertices of a regular pentagon ABCDE with position vectors \(\vec{a}, \vec{a} + \vec{b}, \vec{b}, \lambda \vec{a}, \lambda \vec{b}\) respectively. The ratio \(\frac{AD}{BC}\) is equal to</p>
<p>(a) \(1 - \cos\frac{3\pi}{5} : \cos\frac{3\pi}{5}\)</p>
<p>(b) \(1 + 2\cos\frac{2\pi}{5} : \cos\frac{\pi}{5}\)</p>
<p>(c) \(1 + 2\cos\frac{\pi}{5} : 2\cos\frac{\pi}{5}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the geometry of a regular pentagon and vector magnitudes to establish the relationship between sides and diagonals.
Solution: For a regular pentagon, using the given position vectors and properties of regular pentagons, the ratio \(\frac{AD}{BC} = \frac{1 + 2\cos\frac{\pi}{5}}{2\cos\frac{\pi}{5}}\).
Correct Answer: C