Let $$\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j$$ and $$$\overrightarrow B \left( t \right) = {g_1}\left( t \right)\overrightarrow i + {g_2}\left( t \right)\widehat j,t \in \left[ {0,1} \right],$$$
where $${f_1},{f_2},{g_1},{g_2}$$ are continuous functions. If $$\overrightarrow A \left( t \right)$$ and $$\overrightarrow B \left( t \right)$$ are nonzero vectors for all $$t$$ and $$\overrightarrow A \left( 0 \right) = 2\widehat i + 3\widehat j,$$ $$\,\overrightarrow A \left( 1 \right) = 6\widehat i + 2\widehat j,$$ $$\,\overrightarrow B \left( 0 \right) = 3\widehat i + 2\widehat j$$ and $$\,\overrightarrow B \left( 1 \right) = 2\widehat i + 6\widehat j.$$ Then show that $$\,\overrightarrow A \left( t \right)$$ and $$\,\overrightarrow B \left( t \right)$$ are parallel for some $$t.$$