Side length \ (1 \) square \ (ABCD \) vertex \ (A, D \) respectively \ (X \) axis, \ (Y \) slide on the upside of the shaft, the \ (\ overrightarrow {OB} \ cdot \ overrightarrow {OC} \) the maximum value of \ (\ underline {\ qquad \ qquad} \) .
Analysis:
as shown, denoted \ (E, F \) are \ (AD, BC \) midpoint,
Thus
\ [\ begin {split} \ overrightarrow {OB} \ cdot \ overrightarrow {OC} & = \ left (\ overrightarrow {OF} + \ overrightarrow {FC} \ right) \ cdot \ left (\ overrightarrow {OF} + \ overrightarrow {FB} \ right) \\ & = | OF | ^ 2- | FB | ^ 2 \\ & = | OF | ^ 2- \ dfrac {1} {4} \\ & \ leqslant \ left (| OE | + | EF | \ right
) ^ 2- \ dfrac14 \\ & = 2. \ end {split} \] if and only if \ (E \) point is \ (. OF \) when the line segment, taking the above inequality and the like, the maximum value of the expression is required \ (2 \) .