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 \[\overset{\to }{\mathop{\text{C}}}\,+n\overset{\to
}{\mathop{\text{B}}}\,\].
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