题目内容
若等边△ABC的边长为2，平面内一点M满足
=
+
，则
•
= ．
| CM |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| MA |
| MB |
考点：平面向量数量积的运算
专题：平面向量及应用
分析：由等边△ABC的边长为2，可得
•
=2×2×cos60°．由
=
+
，可得
-
=
+
，
-
=
+
，进而得到
=-
+
，
=
-
．即可得出
•
=(
-
)•(
-
)．
| CA |
| CB |
| CM |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| AM |
| AC |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| BM |
| BC |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| MA |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| MB |
| 2 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| MA |
| MB |
| 1 |
| 2 |
| CA |
| 1 |
| 3 |
| CB |
| 2 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
解答：
解：∵等边△ABC的边长为2，∴CA=CB=2，
•
=2×2×cos60°=2．
∵
=
+
，∴
-
=
+
，
-
=
+
，
∴
=-
+
，
=
-
．
∴
•
=(
-
)•(
-
)=
•
-
2-
2
=
×2-
×22-
×22
=-
．
故答案为：-
．
| CA |
| CB |
∵
| CM |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| AM |
| AC |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| BM |
| BC |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
∴
| MA |
| 1 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| MB |
| 2 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
∴
| MA |
| MB |
| 1 |
| 2 |
| CA |
| 1 |
| 3 |
| CB |
| 2 |
| 3 |
| CB |
| 1 |
| 2 |
| CA |
| 1 |
| 2 |
| CA |
| CB |
| 1 |
| 4 |
| CA |
| 2 |
| 9 |
| CB |
=
| 1 |
| 2 |
| 1 |
| 4 |
| 2 |
| 9 |
=-
| 8 |
| 9 |
故答案为：-
| 8 |
| 9 |
点评：本题考查了数量积的运算及其性质，考查了推理能力和计算能力，属于中档题．
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