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  • please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22 point 4 question 5 maths textbook solution

please solve RD sharma class 12 chapter 22 Algebra of vector exercise 22 point 4 question 5 maths textbook solution

Answers (1)

We need to prove that,

\vec{PA}+\vec{PB}+\vec{PC}+\vec{PD}=4 \vec{PQ}

Hint:

With the help of vector algebra.

Given:

We have given plane ABCD, where Q is point of intersection of the line joining the midpoints of AB and CD; BC and AD.

Solution :

Let \vec{a},\vec{b},\vec{c},\vec{d} be the P.V of the point A,B,C,D respectively.

midpoint of AB=\frac{\vec{a}+\vec{b}}{2}

midpoint of BC=\frac{\vec{b}+\vec{c}}{2}

midpoint of  CD=\frac{\vec{c}+\vec{d}}{2}

midpoint of DA=\frac{\vec{d}+\vec{a}}{2}

 is the midpoint joining the midpoint of AB and CD

Let P be the P.V of P

P.V of Q=\frac{\frac{a-b}{2}+\frac{c+d}{2}}{2}     [ Using midpoint formula \frac{x_{1}+x_{2}}{2}]

=\frac{a+b+c+d}{4}

Let P be the P.V of p

\begin{aligned} &P A+P B+P C+P D=\frac{1}{2}[\vec{a}-\vec{p}+\vec{b}-\vec{p}+\vec{c}-\vec{p}+\vec{d}-\vec{p}] \\ &=(\vec{a}+\vec{b}+\vec{c}+\vec{d})-4 \vec{p} \\ &=4\left(\frac{\vec{a}+\vec{b}+\vec{c}+\vec{d}}{4}-p\right) \\ &=4(\overrightarrow{O Q}-\overrightarrow{O P}) \\ &=4 \overrightarrow{P Q} \end{aligned}.......hence Proved

 

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