question_answer

If \[\overset{\to }{\mathop{A}}\,\times \overset{\to }{\mathop{B}}\,=\overset{\to }{\mathop{C}}\,\times \overset{\to }{\mathop{B}}\,\] , show that \[\overset{\to }{\mathop{C}}\,\] need not be equal to \[\overset{\to }{\mathop{\text{A}}}\,\].

Answer:

                \[\overset{\to }{\mathop{A}}\,\times \overset{\to }{\mathop{B}}\,=\overset{\to }{\mathop{C}}\,\times \overset{\to }{\mathop{B}}\,\] or \[\overset{\to }{\mathop{A}}\,\times \overset{\to }{\mathop{B}}\,\,-\overset{\to }{\mathop{C}}\,\times \overset{\to }{\mathop{B}}\,=0\] or \[(\overset{\to }{\mathop{A}}\,\,-\overset{\to }{\mathop{C}}\,)\times \overset{\to }{\mathop{B}}\,=0\]                                                 ?.. (i) To satisfy (i), the three possibilities can be there (i)\[\overset{\to }{\mathop{\text{A}}}\,-\overset{\to }{\mathop{\text{C}}}\,=0\] or       \[\overset{\to }{\mathop{\text{A}}}\,=\overset{\to }{\mathop{\text{C}}}\,\]                   (ii) \[\overset{\to }{\mathop{\text{B}}}\,=0\] (iii) \[\overset{\to }{\mathop{\text{A}}}\,-\overset{\to }{\mathop{\text{C}}}\,\]and \[\overset{\to }{\mathop{\text{B}}}\,\]are parallel to each other i.e., \[\overset{\to }{\mathop{\text{A}}}\,-\overset{\to }{\mathop{\text{C}}}\,=n\overset{\to }{\mathop{\text{B}}}\,\] , where n is a non zero real number. or\[\overset{\to }{\mathop{\text{A}}}\,=\overset{\to }{\mathop{\text{C}}}\,+n\overset{\to }{\mathop{\text{B}}}\,\] Thus, if \[\overset{\to }{\mathop{\text{A}}}\,\times \overset{\to }{\mathop{\text{B}}}\,=\overset{\to }{\mathop{C}}\,\times \overset{\to }{\mathop{B}}\,\], \[\overset{\to }{\mathop{C}}\,\]need not be equal to \[\overset{\to }{\mathop{\text{A}}}\,\]. The given statement is true if\[\overset{\to }{\mathop{B}}\,\] is a zero vector or \[\overset{\to }{\mathop{\text{A}}}\,\]is equal to \[\overset{\to }{\mathop{\text{C}}}\,+n\overset{\to }{\mathop{\text{B}}}\,\].