Answer:
Given
\[\overset{\to }{\mathop{\text{A}}}\,.\overset{\to }{\mathop{\text{B}}}\,=\overset{\to
}{\mathop{\text{A}}}\,.\overset{\to }{\mathop{C}}\,\], i.e.,
\[AB\,\cos {{\text{ }\!\!\theta\!\!\text{ }}_{\text{1}}}=AC\ \cos {{\text{
}\!\!\theta\!\!\text{ }}_{\text{2}}}\] ??
(i)
Where \[{{\text{ }\!\!\theta\!\!\text{ }}_{\text{1}}}\]is
smaller angle between \[\overset{\to }{\mathop{\text{A}}}\,\]and\[\overset{\to
}{\mathop{B}}\,\]; and\[{{\text{ }\!\!\theta\!\!\text{ }}_{\text{2}}}\] is the
smaller angle between \[\overset{\to }{\mathop{\text{A}}}\,\]and\[\overset{\to
}{\mathop{C}}\,\] .
From (i), \[\text{B}\,\cos {{\text{
}\!\!\theta\!\!\text{ }}_{\text{1}}}=\text{C}\ \cos {{\text{
}\!\!\theta\!\!\text{ }}_{\text{2}}}\] ??
(ii)
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{\text{1}}}={{\text{ }\!\!\theta\!\!\text{
}}_{\text{2}}}\], then from (ii), \[\text{B}\,=C\]or
\[\overset{\to }{\mathop{\text{B}}}\,\,=\overset{\to }{\mathop{\text{C}}}\,\]
If
\[{{\text{ }\!\!\theta\!\!\text{ }}_{\text{1}}}\ne {{\text{
}\!\!\theta\!\!\text{ }}_{\text{2}}}\],then from
(ii), \[\text{B}\,\ne C\] or \[\overset{\to
}{\mathop{\text{B}}}\,\,\,\ne \overset{\to }{\mathop{C}}\,\]
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