Question

If ABCDEF is a regular hexagon and \(\overline{AB}+\overline{AC}+\overline{AD}+\overline{AE}+\overline{AF} = p\overline{AD} = q\overline{AO}\), where O is the center of the hexagon, then the values of \(p\) and \(q\) respectively are

• A. \(2,3\)
• B. \(4,6\)
• C. \(3,6\)
• D. \(3,5\)

Hint:

Add the five vectors from A using position vectors from the centre; the six vertices sum to zero.

Answer 1

The Correct Option is C

Step 1: Use symmetry:
In a regular hexagon, the diagonal AD passes through the centre O, so $\vec{AD} = 2\vec{AO}$. We will write every vector from A using position vectors measured from O.

Step 2: Vector from the centre:
Let position vectors from O be $\vec a, \vec b, \ldots, \vec f$ for A to F. The six vertices sum to zero, $\vec a + \vec b + \vec c + \vec d + \vec e + \vec f = \vec 0$.

Step 3: Sum of the five vectors:
$\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = (\vec b + \vec c + \vec d + \vec e + \vec f) - 5\vec a = (-\vec a) - 5\vec a = -6\vec a = 6\vec{AO}$.

Step 4: Find p:
Since $\vec{AD} = 2\vec{AO}$, the sum is $6\vec{AO} = 3\vec{AD}$. So $p = 3$ and $q = 6$.

Final Answer:
$p = 3, q = 6$. \[ \boxed{p = 3,\ q = 6} \]