如图,在直三棱柱ABC-A1B1C1中,AB=1,AC=AA1=3,BC=2. (1)证明:AB⊥A1C; (2)求二面角A-A1C-B的余弦值.

问题描述:

如图,在直三棱柱ABC-A1B1C1中,AB=1,AC=AA1=

3
,BC=2.

(1)证明:AB⊥A1C;
(2)求二面角A-A1C-B的余弦值.

(1)证明:由直棱柱的性质可得,AA1⊥平面ABC∴AA1⊥AB∵在△ABC中AB=1,AC=3,BC=2,AB2+AC2=BC2∴AB⊥AC又AC∩AA1=A∴AB⊥平面ACC1A1,又∵A1C⊂平面ACC1A1∴AB⊥A1C(2)连接A1C,A1B由已知可得A1B=BC=2 ...