question_answer 47)

                  For two vectors \[\vec{A}\] and \[\vec{B},\,\,|\vec{A}+\,\vec{B}|\]      \[=|\vec{A}-\vec{B}|\] is always true when                 (a)  \[=|\vec{A}|\,=\,|\vec{B}|\,\ne \,0\]                 (b)  \[\vec{A}\,\,\bot \,\,\vec{B}\] (c) \[|\vec{A}|\,=\,|\vec{B}|\,\,\ne \,0\] and A and B are parallel or anti parallel                 (d) when either \[|\vec{A}|\] or \[\,|\vec{B}|\] is zero.

Answer:

                  (b, d) \[|\vec{A}+\vec{B}{{|}^{2}}\,=\,|\vec{A}-\vec{B}{{|}^{2}}\]                 \[\Rightarrow \,(\vec{A}+\vec{B})\,.\,(\vec{A}+\vec{B})\,=\,(\vec{A}-\vec{B})\,.\,(\vec{A}-\vec{B})\]                 or \[4\,\vec{A}.\,\vec{B}=0\] or \[\vec{A}.\vec{B}=0\]                 or \[AB\cos \theta =0\] if \[\theta ={{90}^{o}}\] or either \[|\vec{A}|\,=0\] or \[\,|\vec{B}|\,=0\]