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  • please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22.4 question 2 maths textbook solution

please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22.4 question 2 maths textbook solution

Answers (1)

Hence it has been proved that the sum of median vectors is zero.

Hint:

With the help of vector algebra.

Given:

A, B and C are the three vertices of the triangle and D,E,F are the mid points of the line BD, CA and AB respectively.

Solution :

Let ABC is a triangle such that p.v of  A,B,C are \vec{a},\vec{b} and \vec{c} and c respectively.

As AD,BE and CF are medians ,D,E,F are midpoints.

P.V of D=\frac{\vec{b}+\vec{c}}{2}   [  Using midpoint formula \frac{x_{1}+x_{2}}{2} ]

P.V of D=\frac{\vec{c}+\vec{a}}{2}   [ Using midpoint formula \frac{x_{1}+x_{2}}{2} ]

P.V of D=\frac{\vec{a}+\vec{b}}{2}  [ Using midpoint formula \frac{x_{1}+x_{2}}{2} ]

Now,  \begin{aligned} &\overrightarrow{A D}+\overrightarrow{B E}+\overrightarrow{C F} \\ \end{aligned}

\begin{aligned} &=\left(\frac{\vec{b}+\vec{c}}{2} -\vec{a}\right)+\left(\frac{\vec{c}+\vec{a}}{2} -\vec{b}\right)+\left(\frac{\vec{a}+\vec{b}}{2} - \vec{c}\right) \\ &=\frac{\vec{b}+\vec{c}-2 \vec{a}+\vec{c}+\vec{a}-2 \vec{b}+\vec{a}+\vec{b}-2 \vec{c}}{2} \\ &=\frac{0}{2} \\ &=\overrightarrow{0}\end{aligned}

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