Position Vectors in a Regular Hexagon
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Q.8. In regular hexagon OABCDE, points A and B have P.V $\vec{a}$, $\vec{b}$ resp. referred to the point O. Find the position vectors of C, D, E in terms of $\vec{a}, \vec{b}$ (3 marks). Given: $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$, $\vec{AB} = \vec{OB} - \vec{OA} = \vec{b} - \vec{a}$. Regular hexagon. M is midpoint of OC. $\vec{OM} = \vec{AB} = \vec{b} - \vec{a}$. $\vec{OC} = 2\vec{OM} = 2(\vec{b} - \vec{a}) = [2\vec{b} - 2\vec{a}]$. $\vec{OP} = \vec{OC} + \vec{CD}$. $\vec{OE} = \vec{OM} + \vec{ME} = \vec{b} - \vec{a} + ( - \vec{a}) = [\vec{b} - 2\vec{a}]$.
This question includes visual content: The image shows a regular hexagon labeled OABCDE. Point O is the origin (center left). Vectors OA and OB are given, represented by a and b. There are arrows indicating direction along the sides. A point M is marked as the midpoint of OC. The figure shows various internal segments connecting vertices and the center/midpoints, with vectors labeled in terms of a and b (e.g., b-a, -a, etc.).
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Hi Tanishka, let's solve this regular hexagon vector problem step by step.
Understanding the Figure
We are given a regular hexagon $OABCDE$. The position vectors of points $A$ and $B$ are $\vec a$ and $\vec b$ respectively, measured from point $O$.
Since the hexagon is regular, all sides are equal and parallel sides have the same direction. We will use vector addition carefully to express the remaining vertices.
First, let's find the side vector from A to B. Using position vectors, vector A B equals vector O B minus vector O A.
Now we will determine the position vector of point C.
Finding the Position Vector of C
In a regular hexagon, the line from O to C passes through the center, and the midpoint of O C lies directly above side A B. This means vector O M is equal and parallel to vector A B.
Since M is the midpoint of O C, the full vector O C is twice vector O M.
Substituting the value of vector O M gives us the position vector of C.
Next, let's determine the position vector of point D.
Finding the Position Vector of D
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