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  • Need solution for RD Sharma maths class 12 chapter Scaler and Dot Product exercise 23.2 question 7

Need solution for RD Sharma maths class 12 chapter Scaler and Dot Product exercise 23.2 question 7

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Answer: |\overrightarrow{A B}|=|\overrightarrow{A D}|

Hint: Diagonals of a rectangle are perpendicular

Given: If the rectangle is a square

Solution: Let ABCD be a rectangle.

Suppose the position vectors of points B and D be \vec{a} \text { and } \vec{b}

Now 

\overrightarrow{A C}=\overrightarrow{A D}+\overrightarrow{B C}=\vec{a}+\vec{b}

Also \overrightarrow{ BD}=\vec{a} - \vec{b}

Since ABCD is a rectangle , so \overrightarrow{ AB}\perp \overrightarrow{ AD}

\vec{a} \cdot \vec{b}

Now , diagonals AC and BD are perpendicular if \overrightarrow{ AC}\cdot \overrightarrow{ BD}=0

\begin{aligned} &\text { If }(\vec{a}+\vec{b}) \cdot(\vec{a} \cdot \vec{b})=0 \\ &|\vec{a}|^{2}-|\vec{b}|^{2}=0 \\ &|\vec{a}|^{2}=|\vec{b}|^{2} \\ &|\overrightarrow{A B}|=|\overrightarrow{A D}| \end{aligned}

 

ABCD is a square

 

Thus the diagonal of a rectangle are perpendicular if and only if the rectangle is a square.

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