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Mathematical Methods — Physics STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 SciencePhysicsMathematical Methods3 Marks
Question
If $|\overrightarrow{ A }+\overrightarrow{ B }|=|\overrightarrow{ A }-\overrightarrow{ B }|$ then what can be the angle between $\overrightarrow{ A }$ and $\overrightarrow{ B }$ ?
✓

Answer

Let $\theta$ be the angle between $\overrightarrow{ A }$ and $\overrightarrow{ B }$, then $|\vec{A}+\vec{B}|^2=A^2+B^2+2 A B \cos \theta$
Also the angle between $\vec{A}$ and $-\vec{B}$ is $\left(180^{\circ}-\theta\right)$
Hence $|\vec{A}+\vec{B}|^2=A^2+B^2+2 A B \cos \left(180^{\circ}-\theta\right)$
$=A^2+B^2-2 A B \cos \theta$
As $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$, we can equate above 'two equations, $2 AB \cos \theta=-2 AB \cos \theta$
$ → 4 AB \cos \theta=0$
Assuming $\vec{A}$ and $\vec{B}$ as non-zero vector, we get, $\cos \theta=0$
$→ \theta=90^{\circ}$
Thus, if $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$, then vectors $\vec{A}$ and $\vec{B}$ must be at right angles to each other.

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