Question

The sum of two vectors $\vec{A}$ and $\vec{B}$ is at right angles to their difference. Then

• A. $A=B$
• B. $A=2B$
• C. $B=2A$
• D. $\vec{A}$ and $\vec{B}$ have the same direction

Answer 1

Answer: (A)

The sum of two vectors is $\vec{R}=\vec{A}+\vec{B}$
The difference of two vectors is $\vec{R}=\vec{A}-\vec{B}$
Since $\vec{R}$ and $\vec{R}$ are at right angles, therefore their dot product is zero
i.e., $\vec{R}\cdot \vec{R}=0$ or, $\left(\vec{A}+\vec{B}\right)\cdot \left(\vec{A}-\vec{B}\right)=0$
or $\vec{A}\cdot \vec{A}-\vec{A}\cdot \vec{B}+\vec{B}\cdot \vec{A}-\vec{B}\cdot \vec{B}=0$
or,$A^2-\vec{A}\cdot \vec{B}+\vec{A}\cdot \vec{B}-B^2=0$
(since $\vec{A}\cdot \vec{B}=\vec{B}\cdot \vec{A})$
$\therefore A^2-B^2=0$
$\Rightarrow A=B$