Answer:
Let
\[{{\text{ }\!\!\theta\!\!\text{ }}_{1}}\]be the
smaller angle between \[\overset{\to }{\mathop{\text{A}}}\,\]and \[\overset{\to
}{\mathop{\text{B}}}\,\]; and \[{{\text{
}\!\!\theta\!\!\text{ }}_{\text{2}}}\]be the smaller angle between\[\overset{\to
}{\mathop{\text{A}}}\,\] and \[\overset{\to }{\mathop{\text{C}}}\,\].
Given, \[\overset{\to
}{\mathop{\text{A}}}\,\times \overset{\to }{\mathop{\text{B}}}\,=\overset{\to
}{\mathop{\text{A}}}\,\times \overset{\to }{\mathop{\text{C}}}\,\]
\[\therefore
\] \[AB\sin {{\text{ }\!\!\theta\!\!\text{ }}_{\text{1}}}{{\hat{n}}_{1}}=AC\sin
{{\text{ }\!\!\theta\!\!\text{ }}_{\text{2}}}{{\hat{n}}_{2}}\]
Where\[{{\hat{n}}_{1}}\]
and\[{{\hat{n}}_{2}}\] are unit vectors in the direction of vectors\[(\overset{\to
}{\mathop{\text{A}}}\,\times \overset{\to }{\mathop{\text{B}}}\,)\] and\[(\overset{\to
}{\mathop{\text{A}}}\,\times \overset{\to }{\mathop{\text{C}}}\,)\]
respectively.
From
(i), \[B\sin {{\theta }_{1}}{{\hat{n}}_{1}}=C\sin {{\text{
}\!\!\theta\!\!\text{ }}_{\text{2}}}{{\hat{n}}_{2}}\]
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{1}}={{\text{ }\!\!\theta\!\!\text{
}}_{\text{2}}}\]and\[{{\hat{n}}_{1}}={{\hat{n}}_{2}}\]
, then B = C and \[\overset{\to }{\mathop{\text{B}}}\,=\overset{\to
}{\mathop{\text{C}}}\,\]
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{1}}\ne {{\text{ }\!\!\theta\!\!\text{
}}_{\text{2}}}\]and\[{{\hat{n}}_{1}}\ne
{{\hat{n}}_{2}}\] , then \[\text{B}\ne \text{C}\] and \[\overset{\to }{\mathop{\text{B}}}\,\ne
\overset{\to }{\mathop{\text{C}}}\,\]
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{1}}\ne {{\text{ }\!\!\theta\!\!\text{
}}_{\text{2}}}\]and\[{{\hat{n}}_{1}}={{\hat{n}}_{2}}\]
, then \[\text{B}\ne \text{C}\] and \[\overset{\to }{\mathop{\text{B}}}\,\ne
\overset{\to }{\mathop{\text{C}}}\,\]
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{1}}\text{=}{{\text{ }\!\!\theta\!\!\text{
}}_{\text{2}}}\]and\[{{\hat{n}}_{1}}\ne
{{\hat{n}}_{2}}\] , then B = C and \[\overset{\to }{\mathop{\text{B}}}\,\ne
\overset{\to }{\mathop{\text{C}}}\,\]
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