Vector Algebra
Division of Line Segment
Grade 12
Question:
<p><strong>Ex. 56</strong> In the regular pentagon ABCDE with the given position vectors, AD divides EC in the ratio</p>
<p>(a) \(\cos\frac{2\pi}{5} : 1\)</p>
<p>(b) \(\cos\frac{3\pi}{5} : 1\)</p>
<p>(c) \(1 : 2\cos\frac{\pi}{5}\)</p>
<p>(d) \(1 : 2\)</p>
Step-by-Step Solution
Key Concept: Apply collinearity conditions and the section formula to determine how line AD divides segment EC in a regular pentagon.
Solution: Using the properties of a regular pentagon and applying the section formula with position vectors, the line AD divides EC in the ratio \(1 : 2\cos\frac{\pi}{5}\).
Correct Answer: C