Known three unit vectors $ \ textbf {a}, \ textbf {b}, \ textbf {c} $ satisfy $ \ textbf {a} + \ textbf {b} + \ textbf {c} = \ textbf {0} , \ textbf {e} $ is the unit vector of the plane of any
request $ 2 | \ textbf {e} \ cdot \ textbf {a} | +3 | \ textbf {e} \ cdot \ textbf {b} | +4 | \ textbf {e} \ cdot \ textbf {c} | $ maximum ______
Analysis: Using $ | x | + | y | + | z | = \ max \ {| x + y + z |, | x + yz |, | x-y + z |, | xyz | \} $
readily available $ 2 | \ textbf {e} \ cdot \ textbf {a} | +3 | \ textbf {e} \ cdot \ textbf {b} | +4 | \ textbf {e} \ cdot \ textbf {c} | = \ max \ {\ sqrt {3}, \ sqrt {43}, \ sqrt {39}, \ sqrt {31} \} = \ sqrt {43} $
when $ \ textbf {e} $ and $ 2 \ textbf {a} + 3 \ textbf {b} -4 \ taken when textbf {c} $ collinear.
Exercise: If the real number $ x, y, z \ in [0,1] $ is $ | 3x + 4y-5z | + | 3x-4y + 5z | + | -3x + 4y + 5z | $ _ maximum value of _____
Tip: 15