SATHEE: Vectors 2 Question 24

Vectors 2 Question 24

24. If $A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Then, show that $\sum_{i = 1}^{n - 1} ({\vec{O A}}_{i} \times {\vec{O A}}_{i + 1}) = (1 - n) ({\vec{O A}}_{2} \times {\vec{O A}}_{1})$ .

(1982, 2M)

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Solution:

Since, ${\vec{O A}}_{1}, {\vec{A A}}_{2}, \dots, {\vec{O A}}_{n}$ are all vectors of same magnitude and angle between any two consecutive vectors is same i.e. $(2 π / n)$ .

$∴ {\vec{O A}}_{1} \times {\vec{O A}}_{2} = a^{2} \cdot \sin \frac{2 π}{n} \cdot \hat{p}$

where, $\hat{p}$ is perpendicular to plane of polygon.

Now, $\sum_{i = 1}^{n - 1} (\vec{O A}_{i} \times O \vec{A_{i + 1}}) = \sum_{i = 1}^{n - 1} a^{2} \cdot \sin \frac{2 π}{n} \cdot \hat{p}$

$\begin{aligned} = (n - 1) \cdot a^{2} \cdot \sin \frac{2 π}{n} \cdot \hat{p} \\ = (n - 1) [{\vec{A A}}_{1} \times {\vec{O A}}_{2}] \\ = (1 - n) [{\vec{O A}}_{2} \times {\vec{O A}}_{1}] = R H S \end{aligned}$

Vectors 2 Question 24

24. If $A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Then, show that $\sum_{i = 1}^{n - 1} ({\vec{O A}}_{i} \times {\vec{O A}}_{i + 1}) = (1 - n) ({\vec{O A}}_{2} \times {\vec{O A}}_{1})$ .

Solution:

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Vectors 2 Question 24

24. If A1,A2,…,An are the vertices of a regular plane polygon with n sides and O is its centre. Then, show that ∑i=1n−1(OA→i×OA→i+1)=(1−n)(OA→2×OA→1).

Solution:

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24. If $A_{1}, A_{2}, \dots, A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Then, show that $\sum_{i = 1}^{n - 1} ({\vec{O A}}_{i} \times {\vec{O A}}_{i + 1}) = (1 - n) ({\vec{O A}}_{2} \times {\vec{O A}}_{1})$ .