Question

The sum and difference of two vectors are perpendicular to each other. Prove that the vectors are equal in magnitude.

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Answer 1

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Answer 2

Explanation:

Let x and y be two vectors. It is given that the sum and the difference of two vectors are perpendicular to each other. It means their dot product is equal to 0. So,

$(x+y){\cdot} (x-y)=0\ \because \cos90=0$

So,

$\vec{x}{\cdot} \vec{x}-\vec{x}{\cdot} \vec{y}+\vec{y}{\cdot}\vec{x}-\vec{y}{\cdot} \vec{y}=0$

Since, $\vec{x} {\cdot} \vec{y}=\vec{y} {\cdot} \vec{x}$

So,

$\vec{x}{\cdot} \vec{x}-\vec{x}{\cdot} \vec{y}+\vec{x}{\cdot}\vec{y}-\vec{y}{\cdot} \vec{y}=0\\\\\vec{x}{\cdot} \vec{x}-\vec{y}{\cdot} \vec{y}=0$

Since, $\vec{x}{\cdot} \vec{x}=x^2$

So,

$\vec{x}{\cdot} \vec{x}-\vec{y}{\cdot} \vec{y}=0\\\\x^2-y^2=0\\\\x^2=y^2\\\\x=y$

It is proved that if the sum and difference of two vectors are perpendicular to each other, then the vectors are equal in magnitude.